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IN   MEMORIAM 
FLORIAN  CAJORl 


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INTRODUCTORY  MODERN  GEOMETRY 


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INTRODUCTORY 


MODERN  GEOMETRY 


OF 


POINT,  RAY,  AND  CIRCLE 


BY 


WILLIAM   BENJAMIN  SMITH,  A.M.,  Ph.D.  (Goett.) 

Professor  of  Mathematics  and  Astronomy 
University  of  the  State  of  Missouri 


2lelteftc§  fccroalirt  mit  Sreue, 
j^reunblid^  aufgefafeteS  Jieue. 

—  Goethe. 


WelD  fork 
MACMILLAN   &   CO. 

AND    LONDON 

1893 

All  rights  reserved 


Copyright,  1892, 
By  MACMILLAN  AND  CO. 


Typography  by  J.  S.  Cushing  &  Co.,  Boston,  U.S.A. 
Presswork  by  Berwick  &  Smith,  Boston,  U.S.A. 


g^3 


PREFACE, 


This  book  has  been  written  for  a  very  practical  purpose, 
namely,  to  present  in  simple  and  intelligible  form  a  body  of 
geometric  doctrine,  acquaintance  with  which  may  fairly  be  de- 
manded of  candidates  for  the  Freshman  class,  and  is  in  fact 
demanded  at  this  University  of  the  State  of  Missouri.  This 
purpose  has  regulated  both  the  amount  and  the  character  of  the 
matter  introduced.  The  former  might  have  been  made  larger,  the 
latter  more  uniform  and  scientific,  but  only  —  so  at  least  it  seemed 
to  the  author  —  at  a  sacrifice  of  usefulness  under  existing  con- 
ditions. 

Much  more  than  one  year's  study  can  hardly  be  given  to 
Plane  Geometry  in  the  majority  of  High  Schools  and  Academies 
—  a  fact  that  sets  rather  narrow  limits  to  practicable  treatment 
of  the  subject.  In  such  a  course  the  Apollonian  problem  would 
seem  to  present  itself  as  a  natural  and  rightful  goal.  Besides,  in 
its  solution  the  logical  play,  while  direct  and  simple,  is  yet  highly 
instructive  and  even  artistic,  —  all  the  concepts  of  the  foregoing 
sections  are  summoned  up  and  marshalled  and  brought  to  bear 
upon  a  single  point.  But  if  any  such  goal  is  to  be  attained  in 
such  a  time,  the  path  pursued  must  not  be  tortuous,  and  there 
will  be  little  leisure  for  lateral  excursions.  In  the  Exercises, 
however,  the  view  of  the  student  is  considerably  widened  so  as 
to  embrace  most  of  the  more  familiar  theorems  omitted  from  the 

V 


iM30R2.*?2 


vi  PREFACE. 

text.  These  Exercises  have,  in  fact,  been  cliosen  witli  especial 
reference  not  so  much  to  their  merely  disciplinary  as  to  their 
didactic  value,  the  author  being  persuaded  that  quite  as  good 
exercise  may  be  found  in  going  somewhither  as  in  walking  round 
the  square.  The  problems  proposed  for  solution  will  not  merely 
drill  the  student  in  what  he  already  knows,  but  will  greatly  extend 
his  knowledge,  in  particular,  of  projection  and  perspective,  guid- 
ing him  nearly  as  far  as  he  can  conveniently  go  without  the  help 
of  the  Cross  Ratio  —  a  notion  which  the  narrow  scope  of  the  work 
as  a  mere  introduction  seemed  to  exclude  from  employment.  It 
is  believed  that  advocates  of  the  heuristic  method  may  find  in 
these  problems  ample  playroom  for  the  ingenuity  of  their  pupils. 
As  regards  both  the  matter  and  the  arrangement  of  this  part  of 
the  book,  the  author  would  lay  little  claim  to  originality,  but 
would  rather  acknowledge  indebtedness  to  his  predecessors  in 
the  attempt  to  modernize  geometrical  teaching,  particularly  to 
the  valuable  and  indeed  admirable  works  of  Dupuis,  Halsted, 
Henrici,  Newcomb,  Frischauf,  Henrici  and  Treutlein,  and 
Mueller. 

Up  to  the  Taction-Problem  the  notion  of  Form  has  dominated 
the  whole  discussion,  but  in  the  following  sections  certain  metric 
relations  of  great  importance  receive  due  consideration. 

With  respect  to  the  methods  employed  and  the  point  of  view 
assumed,  a  preface  is  no  place  for  apology.  With  such  as 
approve  the  resolution  of  the  31st  Assembly  of  German  edu- 
cators : 

"  Im  Unterricht  der  Elementargeometrie  an  Realschulen  und  Gymnasien 
bleibt  die  Euclidische  Geometric  dem  System  nach  bestehen,  wird  aber  im 
Geiste  der  neueren  Geometrie  reformiert," 

argument  would  be  needless;  with  others  it  might  be  useless. 
The  case  stands  in  a  measure  as  with  the  Gospel  saying,  made 


PREFACE,  vii 

for  those  who  could  receive  it.  The  work  asks  to  be  judged,  at 
least  in  its  name,  according  to  this  spirit  of  Modern  Geometry, 
and  not  according  to  the  letter. 

In  the  treatment  of  fundamental  notions,  of  Parallels,  of  Pro- 
portion, and  in  fact  throughout  the  book,  the  reader  may  find 
enough  that  is  novel,  if  nothing  that  is  new.  The  author  can- 
not hope  to  escape  criticism,  and  is  himself  aware  of  certain 
defects ;  but  he  may  at  least  trust  that  his  book  may  provoke 
some  abler  pen  to  more  successful  endeavor. 

The  way  of  Mathematics,  it  has  been  said,  is  broad  and 
smooth ;  but  it  is  exceeding  long  and  exceeding  steep.  If  the 
work  in  hand  shall  make  the  first  upward  steps  of  the  climber 
not  indeed  less  difficult,  but  quicker,  longer,  and  less  tedious, 
and  so  conserve  him  time,  energy,  and  disposition  for  much 
higher  ascent,  there  will  be  recompense  for  the  labor  and  even 
for  the  renunciation  that  its  preparation  has  entailed. 

AUTHOR. 

Columbia,  Missouri, 
1st  October,  1892. 

Articles  marked  with  an  asterisk,  *,  may  be  omitted  on  first 
reading.  The  early  attention  of  the  teacher  is  called  to  the  Con- 
cluding Note,  Arts.  353,  sqq. 


CONTENTS. 


Art. 

I-I59 

I-   24 
25,     26 

27-  45 
46-  72 
73-  80 
81-  89 

90-107 

108-139 

140",  140'' 

140-159 


■60-366. 

160-162. 
163-174. 
175-178. 
179-192. 
193-225. 
226-234. 
235-280. 
281-297. 
298-328. 

329-333- 
334-336. 
337-343- 
344-353- 
354-366. 


Page 

Linear  Relations.  1-143 

Introduction 1-20 

Axioms 21,  22 

Congruence 23-  34 

Triangles 34-  56 

Parallelograms 56-  62 

The  4-side,  Parallels,  Concurrents  .     .     .  63-  69 

Exercises  1 69-  76 

Symmetry 76-  90 

The  Circle 90-118 

The  Circle  as  Envelope 119,  120 

Constructions 120-136 

Exercises  II.,  III.,  IV 136-143 

Areal  Relations.  144-292 

Area 144-146 

Criteria  of  Equality 147-154 

Miscellaneous  Applications 154-156 

Squares 157-168 

Proportion 168-187 

Similar  Figures 188-192 

Constructions ,     .     .  192-212 

The  Taction-Problem 212-225 

Metric  Geometry 226-248 

Measurement  of  the  Circle 248-i253 

Measurement  of  Angles 253-255 

The  Euclidian  Doctrine  of  Proportion      .  255-261 

Maxima  and  Minima 261-268 

Concluding  Note 268-274 

Exercises  V 274-292 

Index 293-297 

ix 


GEOMETRY. 


INTRODUCTION. 

1.  Geometry  is  the  Doctrine  of  Space. 

What  is  Space?  On  opening  our  eyes  we  see  objects 
around  us  in  endless  number  and  variety :  the  book  here, 
the  table  there,  the  tree  yonder.  This  vision  of  a  world 
outside  of  us  is  quite  involuntary  —  we  cannot  prevent  it, 
nor  modify  it  in  any  way ;  it  is  called  the  Intuition  (or  Per- 
ception or  Envisagement)  of  Space.  Two  objects  precisely 
alike,  as  two  copies  of  this  book,  so  as  to  be  indistinguishable 
in  every  other  respect,  yet  are  not  the  same,  because  they 
differ  in  place,  in  their  positions  in  Space  :  the  one  is  here, 
the  other  is  not  here,  but  there.  In  between  and  all  about 
these  objects  that  thus  differ  in  place,  there  hes  before  us 
an  apparently  unoccupied  region,  where  it  seems  that  noth- 
ing is,  but  where  anything  might  be.  We  may  imagine  or 
suppose  all  these  objects  to  vanish  or  to  fade  away,  but  we 
cannot  imagine  this  region,  either  where  they  were  or  where 
they  were  not,  to  vanish  or  to  change  in  any  way.  This 
region,  whether  occupied  or  unoccupied,  where  all  these 
objects  are  and  where  countless  others  might  be,  is  called 
Space. 

2 .  There  are  certain  elementary  facts,  that  is,  facts  that 

cannot  be  resolved  into  any  simpler  facts,  about  this  Space, 

and  these  deserve  special  notice. 

1 


2  GEOMETRY. 

A.  Space  is  fixed,  permanent,  unchangeable.  The  objects 
in  Space,  called  bodies,  change  place,  or  may  be  imagined 
to  change  place,  in  all  sorts  of  ways,  without  in  the  least 
affecting  Space  itself.  Animals  move,  that  is,  change  their 
places,  hither  and  thither ;  clouds  form  and  transform  them- 
selves, drifting  before  the  wind,  or  dissolve,  disappearing 
altogether ;  the  stars  circle  eternally  about  the  pole  of  the 
heavens ;  sun,  moon,  and  planets  wander  round  among  the 
stars ;  but  the  blue  dome  of  the  sky,*  the  immeasurable 
expanse  in  which  all  these  motions  go  on,  remains  unmoved 
and  immovable,  as  a  whole  and  in  all  its  parts,  absolutely 
the  same  yesterday,  to-day,  and  forever. 

B.  Space  is  homoeoidal ;  i.e.  it  is  precisely  alike  through- 
out its  whole  extent.  Any  body  may  just  as  well  be  here, 
there,  or  yonder,  so  far  as  Space  is  concerned.  A  mere 
change  of  place  in  nowise  affects  the  Space  in  which  the 
change,  or  motion,  occurs. 

C.  Space  is  boundless.  It  has  no  beginning  and  no  end. 
We  may  imagine  a  piece  of  Space  cut  out  and  colored  (to 
distinguish  it  from  the  rest  of  Space)  ;  the  piece  will  be 
bounded,  but  Space  itself  will  remain  unbounded. 

N.  B.  When  we  say  that  Space  is  unbounded,  we  do  not 
mean  that  it  is  infinite.  Suppose  an  earthquake  to  sink  all 
the  land  beneath  the  level  of  the  sea,  and  suppose  this  latter 
at  rest ;  then  its  outside  would  be  unbounded,  without  begin- 
ning and  without  end,  —  a  fish  might  swim  about  on  it  in 
any  way  forever,  without  stop  or  stay  of  any  kind.  But  it 
would  not  be  infinite ;  there  would  be  exactly  so  many 
square  feet  of  it,  a  finite  number,  neither  more  nor  less. 
Likewise,  the  fact  that  bodies  may  and  do  move  about  in 
space  every  way  without  let  or  hindrance  of  any  kind  implies 

*  Appearing  blue  because  of  the  refraction  of  light  in  the  air. 


INTRODUCTION.  3 

that  Space  is  boundless,  but  by  no  means  that  it  is  infinite. 
For  all  we  know  there  may  be  just  so  many  cubic  feet  of 
Space ;  it  may  be  just  so  many  times  as  large  as  the  sun, 
neither  more  nor  less.  This  distinction  between  unbounded 
and  infinite,  first  clearly  drawn  by  Riemann,  is  fundamental. 

D.  Space  is  continuous.  There  are  no  gaps  nor  holes  in 
it,  where  it  would  be  impossible  for  a  body  to  be.  A  body 
may  move  about  in  Space  anywhere  and  everywhere,  ever 
so  much  or  ever  so  little.  Space  is  itself  simply  where  a 
body  may  be,  and  a  body  may  be  anywhere. 

E.  Space  is  triply  extended,  or  has  three  dimensions. 
This  important  fact  needs  careful  exphcation. 

In  telling  the  size  of  a  box  or  a  beam  we  find  it  necessary 
and  sufficient  to  tell  three  things  about  it :  its  length,  its 
breadth,  and  its  thickness.  These  are  called  its  dimensions  ; 
knowing  them,  we  know  the  size  completely.  But  to  tell  the 
size  of  a  ball  it  is  enough  to  tell  one  thing  about  it,  namely, 
its  diameter ;  while  to  tell  the  size  of  a  chair  we  should 
have  to  tell  many  things  about  it,  and  we  should  be  puzzled 
to  say  what  was  its  length,  or  breadth,  or  thickness.  Never- 
theless, it  remains  true  that  Space  and  all  bodies  in  Space 
have  just  three  dimensions,  but  in  the  sense  now  to  be  made 
clear. 

We  learn  in  Geography  that,  in  order  to  tell  accurately 
where  a  place  is  on  the  outside  of  the  earth,  which  may 
conveniently  be  thought  as  a  level  sheet  of  water,  it  is 
necessary  and  sufficient  to  tell  two  things  about  it ;  namely, 
its  latitude  and  its  longitude.  Many  places  have  the  same 
latitude,  and  many  the  same  longitude ;  but  no  two  have 
the  same  latitude  and  the  same  longitude.  It  is  not  suffi- 
cient, however,  if  we  wish  to  tell  exactly  where  a  thing  is  in 
Space,  to  tell  two  things  about  it.  Thus,  at  this  moment 
the  bright  star  Jupiter  is  shining  exactly  in  the  south ;  we 


4  GEOMETRY. 

also  know  its  altitude,  how  high  it  is  above  the  horizon  (this 
altitude  is  measured  angularly  —  a  term  to  be  explained 
hereafter,  but  with  which  we  have  no  present  concern) .  But 
the  knowledge  of  these  two  facts  merely  enables  us  to  point 
towards  Jupiter ;  they  do  not  fix  his  place  definitely,  they 
do  not  say  how  far  away  he  is  :  we  should  point  towards 
him  the  same  way  whether  he  were  a  mile  or  a  million  of 
miles  distant.  Accordingly,  a  third  thing  must  be  known 
about  him,  in  order  to  know  precisely  where  he  is ;  namely, 
his  distance  from  us.  But  when  this  third  thing  is  known, 
no  further  knowledge  about  his  place  is  either  necessary  or 
possible.  Once  more,  here  is  the  point  of  a  pin.  Where 
is  it  in  this  room  ?  It  is  five  feet  above  the  floor.  This  is 
not  enough,  however,  for  there  are  many  places  five  feet 
above  the  floor.  It  is  also  ten  feet  from  the  south  wall,  but 
there  are  yet  many  positions  five  feet  from  the  floor  and  ten 
feet  from  the  south  wall,  as  we  may  see  by  slipping  a  cane 
five  feet  long  sharpened  to  a  point,  upright  on  the  floor, 
keeping  the  point  always  ten  feet  from  the  south  wall.  But 
as  it  is  thus  slipped  along,  the  point  of  the  cane  will  come 
to  the  point  of  the  pin  and  then  will  be  exactly  twelve  feet 
from  the  west  wall.  If  it  now  move  ever  so  little  either  way 
east  or  west,  it  will  no  longer  be  at  the  pin-point  and  no 
longer  twelve  feet  from  the  west  wall.  So  there  is  one,  and 
only  one,  point  that  is  five  feet  from  the  floor,  ten  feet  from 
the  south  wall,  and  twelve  feet  from  the  west  wall.  Hence 
it  is  seen  that  these  three  facts  fix  the  position  of  the  pin- 
point exactly.  A  fourth  statement,  as  that  the  point  is  nine 
from  the  ceiling,  will  either  be  superfluous,  if  the  ceiling  is 
fourteen  feet  high,  being  implied  in  what  is  already  said,  or 
else  incorrect,  if  the  ceiHng  is  not  fourteen  feet  high,  contra- 
dicting what  is  already  said.  In  general,  with  respect  to  any 
position  in  Space  it  is  necessary  to  know  three  independent 


INTRODUCTION.  5 

facts  (or  data),  and  it  is  impossible  to  know  any  more.  All 
other  knowledge  about  the  position  is  involved  in  tliis  knowl- 
edge, which  is  necessary  and  sufficient  to  enable  us  to  an- 
swer any  rational  question  that  can  be  put  with  respect  to 
the  position.  Accordingly,  since  any  position  in  Space  is 
known  completely  when,  and  only  when,  three  independent 
data  are  known  about  it,  we  say  that  Space  is  triply  or  three- 
fold extended,  or  has  three  dimensions.  The  dimensions  are 
any  three  independent  things  that  it  is  necessary  and  suffi- 
cient to  know  about  any  position  in  Space,  as  of  the  pin- 
point or  of  Jupiter,  in  order  to  know  exactly  where  it  is. 

3.  But  with  respect  to  the  outside  of  the  earth,  viewed 
as  a  level  sheet  of  water,  we  have  seen  that  only  two  data, 
as  of  latitude  and  longitude,  are  necessary  and  sufficient 
to  fix  any  position  on  it ;  neither  are  more  than  two  inde- 
pendent data  possible  ;  all  other  knowledge  about  the  posi- 
tion is  involved  in  the  knowledge  of  these  two  data  about 
it.  Accordingly  we  say  of  such  outside  of  the  earth  that  it 
is  doubly  or  two-fold  extended,  is  bi-dimensional,  or  has  two 
dimensions ;  and  we  name  every  such  outside,  every  such 
bi-dimensional  region,  a  surface.  Such  is  the  top  of  the 
table  :  to  know  where  a  spot  is  on  it  we  need  know  two,  and 
only  two,  independent  facts  about  it,  as  how  far  it  is  from 
the  one  edge  and  how  far  from  the  other.  (Which  other? 
and  why  ?) 

We  see  at  once  that  a  surface  is  no  part  of  Space,  but  is 
only  a  border  (doubly  extended)  between  two  parts  of  Space. 
Thus,  the  whole  earth-surface  is  no  part  either  of  the  earth- 
space  or  of  the  air-space  around  the  earth,  but  is  the  boun- 
dary between  them.  A  soap-bubble  floating  in  the  air  is 
not  a  surface ;  though  exceedingly  thin,  it  has  some  thick- 
ness and  occupies  a  part  of  space ;  the  outside  of  the  film 


6  GEOMETRY. 

is  a  surface,  and  so  is  the  inside,  and  these  are  kept  apart 
by  the  film  itself.  If  the  film  had  no  thickness,  the  outside 
and  the  inside  would  fall  together,  and  the  film  would  be  a 
surface ;  namely,  the  outside  of  the  Space  within  and  the 
inside  of  the  Space  without. 

4.  Consider  now  once  more  this  earth- surface,  still  viewed 
as  a  smooth  level  sheet  of  water.  From  Geography  we 
learn  that  there  are  two  extreme  positions  on  this  surface 
that  are  called  poles  and  that  do  not  move  at  all  as  the 
earth  spins  round  on  her  axis.  We  also  learn  that  there  is 
a  certain  region  of  positions  just  midway  between  these 
poles  and  called  the  Equator.  This  Equator  is  no  part  of 
the  surface ;  it  is  only  a  border  or  boundary  between  two 
parts  of  the  globe-surface,  which  are  called  hemispheres. 
To  know  where,  any  position  is  on  this  border,  it  is  neces- 
sary and  sufficient  to  know  one  thing  about  it,  namely,  its 
longitude ;  neither  is  any  other  independent  knowledge 
about  the  position  possible  ;  all  other  knowledge  is  involved 
in  this  one  knowledge.  Accordingly  we  say  of  this  border, 
the  Equator,  that  it  is  simply  extended,  or  has  one  dimension 
only.  Every  such  one-dimensional  border  is  called  a  line, 
and  its  one  dimension  is  named  length.  A  line,  then,  has 
length,  but  neither  breadth  nor  thickness. 

5.  Lastly,  consider  a  part  of  a  line,  as  of  the  Equator, 
say  between  longitudes  40°  and  50°.  The  ends  of  this  part 
bound  it  off  from  the  rest  of  the  equator,  but  they  them- 
selves form  no  part  of  the  Equator.  They  are  called  points  ; 
they  have  position  merely,  but  no  extent  of  any  kind,  neither 
length  nor  breadth  nor  thickness,  —  they  are  wholly  non- 
dimensional. 


INTRODUCTION.  7 

*6.  It  is  noteworthy  that  extents,  or  regions,  are  bounded 
by  extents  of  fewer  dimensions,  and  themselves  bound 
extents  of  more  dimensions.  Thus,  lines  are  bounded  by 
points,  and  themselves  bound  surfaces  ;  surfaces  are  bounded 
by  lines,  and  themselves  bound  spaces  ;  spaces  are  bounded 
by  surfaces,  and  themselves  bound  —  what  ?  If  anything 
at  all,  it  must  be  some  extent  of  still  higher  order,  oi  four 
dimensions.  But  here  it  is  that  our  intuition  fails  us ;  our 
vision  of  the  world  knows  nothing  of  any  fourth  dimension, 
but  is  confined  to  three  dimensions.  If  there  be  any  such 
fourth  dimension,  we  can  know  nothing  of  it  by  intuition  : 
we  cannot  imagine  it.  In  music,  however,  we  do  recognize 
four  dimensions  :  in  order  to  know  a  note  completely,  to 
distinguish  it  from  every  other  note,  we  must  know  four 
things  about  it :  its  pitch,  its  intensity,  its  length,  its  timbre, 
—  how  high  it  is,  how  loud  it  is,  how  long  it  is,  how  rich 
it  is.  While,  then,  extents  of  higher  dimensions  may  be 
unimaginable,  they  are  not  at  all  unreasonable. 

This  doctrine  of  dimensions  is  of  prime  importance,  but 
rather  subtile  ;  let  not  the  student  be  disheartened,  if  at  first 
he  fail  to  master  it. 

6.  We  may  see  and  handle  bodies,  which  occupy  portions 
of  Space ;  but  not  so  surfaces,  lines,  points,  which  occupy 
no  Space,  but  are  merely  regions  in  Space.  Here  we  must 
invoke  the  help  of  the  logical  process  called  abstraction, 
i.e.  withdrawing  attention  from  certain  matters,  disregard- 
ing them,  while  regarding  others.  A  sheet  of  paper  is  not 
a  surface,  but  a  body  occupying  Space.  However  thin,  it 
yet  has  some  thickness.  But  in  thinking  about  it  we  may 
leave  its  thickness  out  of  our  thoughts,  disregard  its  thick- 
ness altogether ;  so  it  becomes  for  our  thought,  though  not 
for  our  senses  or  imagination,  a  surface.     The  like  may  be 


8  GEOMETRY. 

said  of  the  film  of  the  soap-bubble.  Again,  consider  the 
pointer.  It  is  a  body  or  solid,  not  only  long,  but  wide  and 
thick ;  it  occupies  Space.  It  is  neither  line  nor  surface. 
But  we  may,  and  do  often,  disregard  wholly  two  of  its 
dimensions,  and  attend  solely  to  the  fact  that  it  is  long. 
Thus  it  becomes  for  our  thought  a  line,  though  not  for  our 
senses  or  imagination.  So  the  mark  made  with  chalk  or 
ink  or  pencil  is  a  body,  triply  extended  ;  but  we  disregard 
all  but  its  length,  and  it  becomes  for  our  reason  a  li7ie. 
Lastly,  we  make  a  dot  with  pen  or  chalk  or  pencil ;  it  is  a 
body,  tri-dimensional,  occupying  Space.  But  we  may  dis- 
regard all  its  dimensions,  and  attend  solely  to  the  fact  that 
it  has  position,  that  it  is  here,  and  not  there.  So  it  becomes 
in  our  thought  a  point.  By  such  abstraction  the  earth,  the 
sun,  the  stars,  the  planets,  may  all  be  treated  as  points. 


Fig.  I. 

7.  Inasmuch  as  Space  is  continuous,  there  may  also  be 
continuous  surfaces  and  lines ;  and  the  only  surfaces  and 
lines  treated  in  this  book  are  continuous,  without  holes,  gaps, 
rents,  breaks,  or  interruptions  of  any  kind  in  their  extent. 

It  is  important  to  note  that  in  passing  from  any  position  A 


INTRODUCTION.  9 

to  another  B  on  a  continuous  line,  a  moving  point  P  must 
pass  through  a  complete  series  of  intermediate  positions; 
i.e.  there  is  no  position  on  the  line  between  A  and  B  that 
the  point  P  would  not  assume  in  going  from  A  to  B. 
(Fig.  I.) 

*8.  Starting  from  the  notion  of  Space,  we  have  attained 
the  notions  of  surface,  Hne,  and  point,  in  two  ways  :  by  treat- 
ing them  as  borders,  and  by  the  process  of  abstraction.  But 
we  may  reverse  this  order  and  attain  the  notions  of  line, 
surface,  and  solid  or  space  from  the  notion  of  point,  with 
the  help  of  the  notion  of  motion,  thus  :  Let  a  point  be 
defined  as  \i2cvmg  position  without  parts  or  magnitude  of  any 
kind.  Let  it  move  continuously  through  Space  from  the 
position  A  to  the  position  B.  To  know  where  it  is  at  any 
stage  of  its  motion  along  any  definite  path,  it  is  necessary 


Fig.  2. 

and  sufficient  to  know  one  thing;  namely,  how  far  it  is 
from  A.  Hence  its  path  is  a  ^;?^-dimensional  extent,  or 
what  we  call  a  line. 


10  GEOMETRY. 

Now  let  a  line  move  in  any  definite  way  from  any  position 
Q  to  any  other  position  R.  To  know  the  position  of  any 
point  of  its  path,  it  is  necessary  and  sufficient  to  know  two 
things ;  namely,  the  position  of  the  point  on  the  moving  line 
and  the  position  of  the  moving  line  itself;  hence  the  path 
of  the  line  is  a  /z£/<^-dimensional  extent,  which  we  have 
already  named  a  surface.     (Fig.  2.) 

Now  let  a  surface  move  in  any  definite  way  from  any 
position  U  to  any  other  position  V.  To  know  the  position 
of  any  point  on  its  path  it  is  necessary  and  sufficient  to 
know  three  things  about  it ;  namely,  its  position  on  the 
moving  surface  (which,  we  know,  counts  as  two  things)  and 
the  position  of  the  moving  surface  itself.  Hence  the  path 
of  the  surface  is  a  /"/^rff/f-dimensional  extent,  which  we  have 
already  named  a  solid  or  a  part  of  space. 

Now,  if  we  let  a  solid  move,  what  will  its  path  be? 
Naturally  we  should  expect  it  to  be  a  /^z^r-dimensional 
extent,  but  no  such  extent  is  yielded  in  our  experience  by 
any  motion  of  a  solid  —  the  path  of  a  soUd  is  nothing  but 
a  solid.  The  explanation  of  the  apparent  inconsistency  is 
very  simple,  to-wit :  A  piece  of  a  line  traces  out  a  surface 
only  when  it  moves  out  from  the  line  itself,  —  if  one  part 
were  to  slip  round  on  another  part  of  the  same  line,  it  would 
trace  out  no  surface  at  all  as  its  path ;  likewise,  a  piece  of 
a  surface  traces  out  a  solid  as  its  path  only  by  moving  out 
from  the  surface  itself,  —  if  one  part  were  to  slip  round  on 
another  part  of  the  surface,  it  would  trace  out  no  solid  at 
all  as  its  path.  So,  if  a  piece  of  our  space  could  move  out 
from  space  itself,  it  would  trace  out  a  four-fold  extent  as  its 
path ;  in  fact,  however,  no  part  of  space  can  move  out  from 
space ;  on  the  contrary,  it  can  only  slip  along  in  space,  from 
one  part  of  space  to  another,  and  hence  does  not  trace  out 
any  four-fold  extended  path. 


INTRODUCTION.  11 

9.  Space,  we  have  seen,  is  homoeoidal,  everywhere  ahke. 
We  naturally  inquire  :  Is  there  any  homoeoidal  surface  ?  In 
general,  surfaces  are  certainly  7iot  homoeoidal.  Consider  an 
egg-shell,  and  by  abstraction  treat  it  as  a  surface.  It  is  not 
alike  throughout ;  the  ends  are  not  like  each  other,  and 
neither  is  like  the  middle  region.  Suppose  a  piece  cut  out 
anywhere ;  if  slipped  about  over  the  rest  of  the  shell,  this 
piece  will  not  fit.  But  now  consider  a  smooth  round  ball 
covered  with  a  thin  rigid  film,  and  treat  this  film  as  a  sur- 
face, by  disregarding  its  thickness.  Suppose  a  piece  of  the 
film  cut  out  and  slipped  round  over  the  rest  of  the  film  :  the 
piece  will  fit  everywhere  perfectly,  the  surface  is  homoeoidal ; 
it  is  called  a  sphere-surface. 

N.B.  The  precise  definition  of  this  surface  is  that  all 
its  points  are  equidistant  from  a  point  within^  called  the 
centre.  Suppose  a  rigid  bar  of  any  shape,  pointed  at  both 
ends,  and  movable  about  one  end  fixed  at  a  point ;  then 
the  other  end  will  move  always  on  a  sphere-surface,  which 
iis  the  whole  region  where  the  moving  end  may  be.  Since 
Space  is  homoeoidal  around  the  fixed  point,  the  surface 
everywhere  equidistant  from  the  point  is  also  homoeoidal. 

Now  turn  over  the  piece  cut  out  of  this  spherical  film 
and  slip  it  about  the  film  :  it  no  longer  fits  anywhere  at  all 
—  the  surface  is  homoeoidal,  but  not  reversible. 

10.  But  now  consider  a  fine  mirror  covered  with  a  deli- 
cate film,  which  by  abstraction  we  treat  as  a  surface.  Sup- 
pose a  piece  cut  out  of  the  film  and  slipped  about  over  it : 
the  piece  fits  everywhere ;  turn  it  over,  re-apply  it,  and  slip 
it  about :  it  still  fits  everywhere  —  the  surface  is  both  homoe- 
oidal d^n^  reversible  ;  it  is  called  a  plane-surface. 

*  N.B.  A  precise  definition  of  this  surface  is  the  following  : 
Take  two  points  A  and  B  and  suppose  two  equal  spherical 


12  GEOMETRY. 

bubbles  formed  about  A  and  B  as  centres.     Let  them  ex- 
pand, always  equal  to  each  other,  until  they  meet,  and  still 
keep  on  expanding.    The  line  where 
^  S   the  equal  (Fig.  3)  spherical  bubbles, 

^^'  ^'  regarded  as  surfaces,  meet,  has  all 

its  points  just  as  far  from  A  as  from  B.  As  the  bubbles 
still  expand,  this  line,  with  all  its  points  equidistant  from  A 
and  B,  itself  expands  and  traces  out  a  plane  as  its  path 
through  Space. 

Hence  we  may  define  \ki^  plane  as  the  region  (or  surface) 
where  a  point  may  be  that  is  equidistant  from  two  fixed 
points.  Instead  of  region  it  is  common  to  say  locus,  i.e. 
place.  Briefly,  then,  a  plane  is  the  locus  of  a  point  equidis- 
tant from  two  fixed  points.  It  is  evident  that  the  plane,  as 
thus  defined,  is  reversible ;  for  since  the  bubbles  about  A 
and  B  are  all  the  time  precisely  equal,  to  exchange  A  and 
B,  or  to  exchange  the  sides  of  the  plane,  will  make  no  dif- 
ference whatever.  Thus  the  plane  cuts  the  Space  evenly 
half  in  two ;  and  since  Space  itself  is  homoeoidal,  so  also  is 
this  section  or  surface  that  halves  it  exactly.  The  superiority 
of  this  definition  consists  in  its  not  only  telling  what  surface 
the  plane  is,  but  also  making  clear  that  there  actually  is  such 
a  surface. 

11.  The  mirror  is  the  nearest  approach  that  we  can  make 
to  a  perfect  plane  surface ;  the  blackboard  is  not  plane,  it 
is  rough  and  warped ;  but  we  shall  disregard  all  its  uneven- 
ness  and  treat  it  as  a  plane  extended  through  Space  without 
end.  Any  surface  may  be  dealt  with  as  a  plane  by  abstrac- 
tion, being  thought  as  hommoidal  and  reversible. 

12.  On  this  board,  regarded  as  a  plane,  we  draw  a  chalk- 
mark,  abstract  from  all  its  dimensions  but  its  length,  and 


INTRODUCTION. 


13 


treat  it  as  a  line.     This  line  is  plainly  not  alike  throughout ; 
a  piece  cut  out  and  slipped  along  it  will  not  fit  (Fig.  4). 


But  here  is  a  line  homceoidal,  alike  in  all  its  parts ;  it  is 
drawn  with  a  pair  of  compasses  and  is  called  a  circle 
(Fig.  5).  One  point  of  the  compasses  is  held  fast  at  the 
centre  O,  while  the  other  traces  out  the  circle  as  its  path  in 


S 


Fig.  5. 


Fig.  6. 


the  plane.  The  circle  is  the  locus  of  a  point  in  the  plane 
equidistant  from  a  fixed  point.  Since  the  plane  is  homce- 
oidal, so  too  is  this  circle  (see  Art.  10)  ;  a  piece,  called  an 
arc,  cut  out  and  slipped  round  will  everywhere  fit  on  the 
circle.  But  turn  it  over  and  slip  it  round,  —  it  fits  nowhere ; 
the  circle  is  not  reversible.     It  divides  the  plane  into  two 


14  GEOMETRY. 

parts,  not  halves,  that  are  not  ahke  along  the  dividing  line. 
But  now  suppose  a  perfectly  flexible  string  fastened  at  6"  and 
stretched  by  a  weight  W.  Its  length  only  being  regarded, 
it  is  a  line  homoeoidal,  alike  throughout,  and  also  reversible ; 
any  part  AB  will  not  only  fit  perfectly  anywhere  on  it,  but 
will  also  fit  when  reversed,  turned  end  for  end.  Such  a 
line  is  called  right,  or  straight,  or  direct,  or  a  ray.  Extended 
indefinitely,  it  cuts  the  whole  plane  into  two  halves  pre- 
cisely alike  along  the  ray  itself. 

*N.B.  The  common  hne  where  the  two  spherical  bub- 
bles of  Art.  lo  meet  is  a  circle,  for  it  is  plainly  precisely 
alike  all  around ;  it  is  homoeoidal,  being  the  intersection  of 
two  homoeoidal  surfaces,  namely,  the  two  equal  sphere- 
surfaces  ;  it  is  also  in  a  plane,  and  in  fact  traces  out  the 
plane  by  its  expansion  as  the  bubbles  expand. 

To  get  accurately  the  notion  of  the  ray  or  straight  line, 
we  need  another  point  C,  and  a  third  expanding  bubble 
always  equal  to  those  about  A  and  B.  The  circular  inter- 
section of  the  bubbles  about  A  and  B  will  trace  out  one 
plane  ;  of  those  about  B  and  C  will  trace  out  another  plane ; 
of  those  about  C  and  A  will  trace  a  third  plane.  All  the 
points  where  the  first  two  planes  intersect  will  be  equidistant 
from  A  and  B  and  C,  and  no  other  points  will  be ;  the 
same  may  be  said  of  all  points  where  the  second  and  third 
planes  meet,  and  of  all  points  where  the  third  and  first  meet ; 
hence  all  three  of  the  planes  meet  together,  and  they  meet 
only  together.  Also,  the  line  where  they  meet  has  every 
one  of  its  points  equidistant  from  all  the  three  points,  A,  B, 
C ;  hence  it  is  the  locus  o/  a  point  equidistant  fro ju  three 
fixed  points.  Moreover,  it  is  homoeoidal  and  reversible, 
since  it  is  the  intersection  of  two  planes,  which  are  homoe- 
oidal and  reversible  ;  hence  it  is  what  we  call  a  straight 
line,  or  right  line,  or  ray. 


INTRODUCTION.  15 

13.  We  may  now  define  : 

A  sphere- surface  is  the  locus  of  a  point  at  a  fixed 
distance  from  a  fixed  point.  It  is  homoeoidal,  but  not 
reversible. 

A  plane  is  the  locus  of  a  point  equidistant  from  two  fixed 
points.     It  is  both  homoeoidal  and  reversible. 

A  ray  is  the  locus  of  a  point  equidistant  from  three  fixed 
points.  It  is  both  homoeoidal  and  reversible  ;  it  is  also  the 
intersection  of  two  planes. 

A  circle  is  the  locus  (or  path)  of  a  point  in  a  plane  at  a 
fixed  distance  from  a  fixed  point.  It  is  homoeoidal,  but  not 
reversible.  It  is  also  the  locus  (or  path)  of  a  point  in  space 
at  a  fixed  distance  from  two  points ;  it  is  also  the  intersec- 
tion of  two  equal  sphere-surfaces.* 

14.  It  is  only  with  the  foregoing  figures  and  combinations 
of  them  that  we  have  to  deal  in  this  book.  Circles  and  rays 
may  be  drawn  with  exceeding  accuracy,  but  any  lines,  how- 
ever roughly  drawn,  may  answer  our  logical  purposes  as 
well  as  the  most  accurately  drawn ;  we  have  only,  by 
abstraction,  to  treat  them  as  having  the  character  of  the 
lines  in  question. 

Circles  and  sphere-surfaces  are  unbounded,  without  be- 
ginning or  end,  but  both  are  finite  :  we  shall  learn  how  to 
measure  them. 


*  In  the  foregoing  free  use  has  been  made  of  the  notion  of  equidistance  without 
formal  definition,  because  of  its  familiarity.  We  may,  however,  say  precisely:  \i  A 
and  B  be  two  points,  the  ends  of  a  rigid  bar  of  any  shape,  and  if  A  be  held  fast,  then 
all  the  points  on  which  B  can  fall  are  equidistant  from  A ,  and  no  other  points  are 
equidistant  with  them.  They  all  lie  on  a  closed  surface,  called  a  sphere-surface. 
All  points  within  this  surface  are  said  to  be  less  distant,  and  all  points  without  are 
said  to  be  more  distant,  from  A  than  B  is.  Herewith,  then,  we  tell  exactly  what 
we  mean  by  equidistant,  less  distant,  and  more  distant,  but  we  make  no  attempt  to 
define  distance  in  general,  which  is  difficult  and  unnecessary  to  our  purpose. 


16  GEOMETRY. 

15.  Any  geometric  element  or  combination  of  geometric 
elements,  as  points,  lines,  surfaces,  is  called  a  geometric 
figure.  It  is  a  fundamental  assumption,  justified  by  experi- 
ence, that  space  is  homoeoidal,  that  figures  or  bodies  are  not 
affected  in  size  or  shape  by  change  of  place.  Two  figures 
that  may  be  fitted  exactly  on  each  other,  or  may  be  thought 
so  fitted,  are  called  congruent.  Any  two  points,  lines,  or 
parts  of  the  two  figures,  that  fall  upon  each  other  in  this 
superposition  are  said  to  correspond.  It  is  manifest  that  all 
planes  are  congruent  and  all  rays  are  congruent.  Rays  and 
planes  are  unbounded,  but  whether  or  not  they  are  finite  is 
a  question  that  we  are  unable  to  answer. 

16.  Any  part  of  a  circle  or  ray,  as  AB,  is  bounded  by 
two  end-points,  A  and  B,  and  is  finite ;  the  one  is  named 
an  arc  (Fig.  5),  the  other  a  tract,  sect,  or  line-segment. 
Each  is  denoted  by  the  two  letters  denoting  the  ends,  as 
the  tract  AB,  the  arc  AB.  Sometimes  it  is  important  to 
distinguish  these  end-points  as  beginning  and  end  proper ; 
we  do  this  by  writing  the  letter  at  the  beginning  first. 


A' 

B' 

A 

B 

C 

D 

E 

F 

A 

B' 

Fig.  7. 

F' 

17.  Two  tracts,  AB  and  A^B\  are  called  equal  when  the 
end-points  of  the  one  may  be  (Fig.  7)  simultaneously  fitted 
on  the  end-points  of  the  other. 

If  we  have  a  number  of  tracts,  AB,  CD,  EF,  etc.,  and 
we  lay  off  successively  on  a  ray  tracts  AB\  CD\  E'F,  etc., 


INTRODUCTION.  17 

respectively  equal  to  AB,  CD,  EF,  etc.,  the  end  of  the  first 
being  the  beginning  of  the  second,  and  so  on,  while  no  part 
of  one  falls  on  any  part  of  another,  we  are  said  to  add  or 
sum  the  tracts  AB,  etc.  Each  is  called  an  addend  or  sum- 
mand,  and  the  whole  tract  from  first  beginning  to  last  end 
is  called  the  sum. 

Equality  is  denoted  by  the  bars  ( =  )  between  the  equals, 
as  AB  =  CD. 

1 8.  If,  when  the  beginning  A  is  placed  on  the  beginning 
C,  the  end  B  does  not  fall  on  the  end  D,  the  tracts  are 
unecjual,  and  we  write  AB  4^  CD.  If  B  falls  between  C 
and  D,  then  AB  is  called  less  than  CD,  AB  <  CD ;  but  if 
D  falls  between  A  and  B,  then  AB  is  called  greater  than 
CD,  AB  >  CD.  In  either  case,  the  tract  BD  or  DB, 
between  the  two  ends  of  the  tracts,  whose  beginnings  coin- 
cide, is  called  the  difference  of  the  two  tracts,  and  we  are 
said  to  subtract  the  one  from  the  other.  Ordinarily  we 
mention  the  greater  tract  first  in  speaking  of  difference. 

19.  The  symbols  of  addition  and  subtraction  are  -\-  and 
—  (plus  and  minus),  thus  : 

AB+  CD=AD  and  AB  -  CD  =  BD. 

It  is  important  to  note  here  the  order  of  the  letters.  In 
summing  a  number  of  tracts,  as  AB,  CD,  EF,  etc.,  to  KL, 


BCD  K 

Fig.  8. 

we  have  AB  +  CD  ^-  EF--  -\-  KL  =  AL  (Fig.  8) .  The 
order  of  the  summands  is  indifferent,  and  this  important 
fact  is  called  the  Commutative  Law  of  Addition.     Thus 

AB^  CD\EF=AB^EF-\-  CD=EF-\-AB-\-  CD,  etc. 


18  GEOMETRY. 

20.  When  beginning  and  end  of  a  tract  or  of  any  mag- 
nitude are  exchanged,  the  tract  or  magnitude  is  said  to  be 
reversed,  and  the  reverse  is  denoted  by  the  sign  — .  Thus 
the  reverse  of  AB  is  BA,  or  AB  =  —  BA.  If  we  add  a 
magnitude  and  its  reverse,  the  sum  is  o,  or 

AB  +  {-AB)  =  AB  +  BA  =  o. 

The  same  result  o  is  obtained  by  subtracting,  from  a  magni- 
tude, itself  or  an  equal  magnitude;  and,  in  general,  it  is 
plain  that  to  subtract  CD  yields  the  same  result  as  to  add 
(Fig.  9)  the  reverse  DC.     The  reverse  of  a  magnitude  is 


A  B 

CD  c 


A  B 

Fig.  9. 

often  called  its  negative,  the  magnitude  itself  being  called 
its  positive. 

Similar  rules  hold  for  adding  and  subtracting  arcs  of  a 
circle  or  of  equal  circles. 

ANGLES. 

21.  The  indefinite  extent  of  a  ray  on  one  side  of  a  point 
O,  as  OA,  is  called  a  half-ray :  it  has  a  beginning  O,  but  no 
end.  Two  half-rays,  OA  and  0A\  which  together  make  up 
a  whole  ray,  are  called  opposite  or  counter  (Fig.  10). 

Now  let  two  half-rays,  OA  and  OB,  have  the  same  be- 
ginning O ;  the  opening  or  spread  between  them  is  a  mag- 
nitude :  it  may  be  greater  or  less.  Suppose  OA  and  OB  to 
be  two  very  fine  needles  pivoted  at  O ;  then  OB  may  fall 
exactly  on  OA,  or  it  may  be  turned  round  from  OA ;  and 


INTRODUCTION.  19 

the  amount  of  turning  from  OA  to  OB^  or  the  spread 
between  the  half-rays,  is  called  the  angle  between  them. 
We  may  denote  it  by  a  Greek  letter,  as  a,  written  in  it ;  or 
by  a  large  Roman  letter,  as  (9,  at  its  vertex  (where  the  half- 
rays  meet)  ;  or  by  three  such  letters,  as  A  OB,  the  middle 


Fig.  io. 

one  being  at  the  vertex,  the  other  two  anywhere  on  the  half- 
rays.     The  symbol  for  angle  is  ^. 

22.  The  angle  is  perfectly  definite  in  size,  it  has  two 
ends  or  boundaries ;  namely,  the  two  half-rays,  sometimes 
called  arms.  When  we  would  distinguish  these  arms  as 
beginning  and  end,  we  mention  the  letter  on  the  beginning- 
arm  first,  and  the  letter  on  the  end-arm  last ;  thus,  AOB ; 
here  OA  is  the  beginning  and  OB  the  end  of  the  angle. 

Exchanging  beginning  and  end  reverses  the  angle ;  thus, 
BOA  =  -AOB. 

23.  Two  angles  whose  ends  or  arms  may  be  made  to  fit 
on  each  other  simultaneously  are  named  equal ;  they  are  also 
congruent.  Two  angles  whose  arms  will  not  fit  on  each  other 
simultaneously  are  unequal;  and  that  is  the  less  angle  whose 
end-arm  falls  within  the  other  angle  when  their  beginnings 


20 


GEOMETRY, 


coincide;    the    other   is  the  greater;    thus,    AOB>  AOC 
(Fig.  II). 

24.  We  sum  angles  precisely  as  we  sum  tracts ;  we  lay 
off  a,  p,  etc.,  around  (9,  making  the  end  of  each  the  begin- 
ning of  the  next :  the  angle  from  first  beginning  to  last  end 


Fig.  II. 

is  the  sum.  So,  too,  in  order  to  subtract  ^  from  a,  lay  off 
/8  from  the  beginning  towards  the  end  of  a ;  the  angle  from 
the  end  of  /?  to  the  end  of  a  is  the  difference,  a  —  (i.  Or 
we  may  add  to  a  the  reverse  (or  negative)  of  /? :  the  sum 
will  be  «  +  (— y8)  or  a  — ^  (Fig.  12).^ 

1  It  is  important  to  note  the  close  correspondence  of  tract  and  angle:  the  former  is 
related  to  points  as  the  latter  is  to  rays  (or  half-rays).  The  tract  is  the  simplest 
magnitude  that  lies  between  points,  that  distinguishes  them  and  keeps  them  apart; 
likewise  the  angle  is  the  simplest  magnitude  that  lies  between  rays  (in  a  plane),  that 
distinguishes  them  and  keeps  them  apart.  So,  too,  we  define  equality  and  inequality 
among  tracts  and  among  angles,  quite  similarly,  and  without  being  compelled  before- 
hand to  form  the  notion  of  the  size  either  of  a  tract  or  of  an  angle.  We  may  now 
define  the  distance  between  two  points  to  be  the  tract  between  them,  and  the  dz's- 
tance  between  two  (half-)  rays  to  be  the  angle  between  them,  leaving  for  future 
decision  which  tract  and  which  angle  if  there  should  prove  to  be  several. 


INTRODUCTION. 
D 


21 


Fig.  12, 


AXIOMS. 

25.  At  this  stage  we  must  recognize  and  use  certain  dic- 
tates or  irresoluble  facts  of  experience,  called  axioms. 
{'k^ni)imv[\G^2in?>  something  worthy,  like  the  Latin  dignitas ;  in 
fact,  older  writers  use  dignity  in  the  sense  of  axiom.  But 
Euclid's  phrase  is  Kotvat  Iwomi  —  common  notions.^  Some 
have  no  special  reference  to  Geometry,  but  pervade  all  of 
our  thinking  about  magnitudes  ;  such  are 

(i)  Things  equal  to  the  same  thing  are  equal  to  each 
other. 

(2)  If  equals  be  added  to,  subtracted  from,  multiplied  by, 
or  divided  by,  equals,  the  results  will  be  equal. 


22  GEOMETRY. 

(3)  If  equals  be  added  to  or  subtracted  from  unequals, 
the  latter  will  remain  unequal  as  before. 

(4)  The  whole  equals,  or  is  the  sum  of,  all  its  distinct 
parts,  and  is  greater  than  any  of  its  parts. 

(5)  If  a  necessary  consequence  of  any  supposition  is 
false,  the  supposition  itself  is  false. 

Others  concern  Geometry  especially,  as  : 

(6)  All  planes  are  congruent. 

(7)  Two  rays  can  meet  in  only  one  point. 

The  extremely  important  axiom  (7)  may  be  stated  in 
other  equivalent  ways,  thus  :  Two  rays  cannot  meet  in  two 
or  more  points ;  or.  Two  rays  cannot  have  two  or  more 
points  in  common ;  or.  Only  one  ray  can  go  through  two 
fixed  points ;  or,  A  ray  is  fixed  by  two  points. 

26.  A  statement  or  declaration  in  words  is  called  -3, prop- 
osition. The  propositions  with  which  we  have  to  deal  state 
geometric  facts  and  are  also  called  Theorems  {Oeoiprjfxa,  from 
OewpcLv,  to  look  at,  means  the  product  of  mental  contempla- 
tion). Propositions  are  often  incorrect;  theorems,  never. 
Subordinate  facts,  special  cases  of  general  facts,  and  facts 
immediately  evidenced  from  some  preceding  facts,  are 
called  Corollaries  or  Porisms  {Tvopiafxa  =  deduction). 

We  may  now  proceed  to  investigate  lines  and  angles,  and 
find  out  what  we  can  about  them.  The  first  and  simplest 
things  we  can  learn  concern 


Th.  1.]  CONGRUENCE,  ?3 

CONGRUENCE. 

27.    Theorem  I.  —  All  rays  are  congruent. 

Proof.  Let  L  and  V  be  any  two  rays  (Fig.  13).  On 
L  take  any  two  points,  A  and  B ',  on  L'  take  any  two 
points,  A'  and  B',  so  that  the  tract  AB  shall  equal  the 
tract  A'B'.  Think  of  Z  and  L'  as  extremely  fine  rigid 
spider-threads,  and  in  thought  place  the  ends  of  the  tract 
AB  on  the  ends  of  the  tract  A'B',  A  on  A',  and  B  on  B'. 


B 


A  B 

Fig.  13. 

Then  A  and  A'  become  one  and  the  same  point,  and  B  and 
B'  become  one  and  the  same  point ;  through  these  two 
points  only  one  ray  can  pass  (by  Axiom  7)  :  hence  Z  and 
Z',  which  go  through  these  two  points,  now  become  one 
and  the  same  ray ;  that  is,  they  fit  precisely,  they  are  con- 
gruent. Quod  erat  demonstrandum  =  which  was  to  be 
proved  =  oTTcp  iSa  Set^at,  —  the  solemn  Greek  formula; 
whereas  the  Hindu,  appealing  directly  to  intuition,  merely 
said  Paqya  —  Behold  ! 

28.  In  the  foregoing  proof  we  assumed  that  on  any  ray 
we  could  lay  off  a  tract  equal  to  a  given  tract,  or  that  on 
any  ray  we  could  find  two  points,  A  and  B^  as  far  apart  as 
two  other  points.  A'  and  B\  This  assumption  that  something 
can  be  done,  is  called  a  Postulate  (atrry/xa),  i.e.  a  demand, 
which  must  be  granted  before  we  can  proceed  further. 
Actually  to  carry  out  the  construction,  we  need  a  pair  of 
compasses. 


24  GEOMETRY.  [Th.  II. 

29.  Theorem  II.  — If  two  points  of  a  ray  lie  in  a  certain 
plane,  all  points  of  the  ray  lie  in  that  plarie. 

Proof.  Regard  the  surface  of  paper  or  of  the  blackboard 
as  a  plane,  and  suppose  it  covered  with  a  fine  rigid  film, 
itself  a  plane.  Let  L  be  any  ray  having  two  points,  A  and 
B,  in  this  plane.  Through  these  two  points  suppose  a 
second  plane  drawn  or  passed;  by  definition  (Art.  13)  it 
will  intersect  our  first  plane,  or  film,  along  a  ray  /;  this  ray 
/  goes  through  the  two  points,  A  and  B,  and  lies  wholly 
(with  all  its  points)  in  the  first  plane  ;  also  the  ray  L  goes 
through  A  and  B,  and  only  one  ray  can  go  through  the  same 
two  points,  A  and  B,  by  Axiom  7  ;  hence  L  and  /  are 
the  same  ray ;  but  /  has  all  its  points  in  the  first  plane ; 
hence  L  has  all  its  points  in  the  first  plane,     q.  e.  d. 

Query  :    What  postulate  is  assumed  in  this  proof  ? 

Corollary.  If  a  ray  turn  about  a  fixed  point  P^  and  glide 
along  a  fixed  ray  Z,  it  will  trace  out  a  plane  (Fig.  14). 


Fig.  14. 


For  it  will  always  have  two  points  —  namely,  the  fixed 
point  and  a  point  on  the  fixed  ray  —  in  the  plane  drawn 
through  the  fixed  point  and  the  fixed  ray. 


Th.  II.]  CONGRUENCE.  25 

Query  :  What  postulate  is  here  implied  ?  —  Henceforth  it 
is  understood  that  all  our  points,  lines,  etc.,  are  complanar, 
i.e.  lie  in  one  and  the  same  plane. 

30.  In  the  foregoing  Theorem  and  Corollary  we  observe 
clauses  introduced  by  the  word  if.  Such  a  clause  is  called 
an  Hypothesis,  i.e.  a  supposition.  The  result  reached  by 
reasoning  from  the  hypothesis  and  stated  immediately  after 
the  hypothesis,  is  called  the  Conclusion. 

31.  All  logical  processes  consist  in  one  or  both  of  two 
things  :  the  formation  of  concepts,  as  of  lines,  surfaces, 
angles,  etc.,  and  the  combination  of  these  concepts  into 
propositions.  Geometric  concepts  are  remarkable  for  their 
perfect  clearness  and  precision  —  we  know  exactly  what  we 
mean  by  them  ;  this  cannot  be  said  of  many  other  concepts, 
about  which  diverse  opinions  prevail,  as  in  Political  Economy. 
Hence  it  is  that  Geometry  offers  an  unequalled  gymnasium 
for  the  reason  or  logical  faculty.  We  shall  now  generate 
some  new  concepts.  Let  the  student  note  their  definiteness 
as  well  as  the  mode  of  their  formation. 

32.  Let  OA  and  OB  be  any  two  co-initial  half- rays, 
forming  the  angle  A  OB.  Think  of  OA  as  held  fast  and  of 
OB  as  turning  about  the  pivot  O,  starting  from  the  position 
OA.     As  it  turns  (counter-clockwise),  the  (Fig.  15)  angle 


A' 


AOB  increases.     Finally,  let  it  return  to  its  original  posi- 
tion, OA  ;  then  the  whole  amount  of  turning  from  the  upper 


26  GEOMETRY.  [Th.  III. 

side  of  OA  back  to  the  under  side  of  OA,  or  the  full  spread 
around  the  point  O,  is  called  a  /////  angle  (or  round  angle, 
or  <r/r^«/«- angle,  or  perigon) .  Think  of  a  fan  opened  until 
the  first  rib  falls  on  the  last.  —  Note  that  the  upper  and 
under  sides  of  OA  are  exactly  the  same  in  position,  and  are 
distinguished  only  in  thought.  (Think  of  a  circular  piece 
of  paper  slit  straight  through  from  the  edge  to  the  centre.) 
The  like  may  be  said  of  the  two  sides  of  any  line  or  surface. 
We  can  now  prove 

33.    Theorem  III.  — All  rotijid  angles  are  congruent. 

Proof.  Let  AOB  and  A'O'B^  (Fig.  16)  be  any  two 
round  angles.  Slip  the  half-ray  OA  down,  and  turn  it  till 
OA  falls  on  0A^\     they  will  fit  perfectly  (why?);     the 


Fig.  16. 

whole  round  angle  about  O  will  fit  perfectly  on  the  whole 
round  angle  about  O'  (why  ?)  ;  hence  the  two  full  angles 
are  congruent.     Q.  e.  d. 

N.B.  In  this  sUpping  of  figures  about  in  the  plane,  it  is 
well  to  imagine  the  plane  to  consist  of  two  very  thin,  per- 
fectly rigid,  smooth  and  transparent  films ;  also,  to  imagine 
one  figure  drawn  in  the  lower  film  and  one  in  the  upper ; 
and  to  imagine  the  upper  slipped  about  at  will  over  the  lower. 

Query :  On  what  cardinal  property  of  the  plane  do  these 
considerations  hinge  ? 


Th.  IV.] 


CONGRUENCE, 


27 


34.  From  O  draw  any  half-ray  OA ;  then  any  second 
half- ray  from  O,  as  OB,  will  (Fig.  17)  cut  the  round  angle 
AOA  into  two  angles,  AOB  and  BOA.  The  end  OB  of 
the  first  falls  on  the  beginning,  OB,  of  the  second ;  while 

A 
B 


Fig.  17. 

the  end,  OA,  of  the  second  falls  on  the  beginning,  OA,  of 
the  first.  Hence  the  round  angle  AOA  is  their  sum,  by 
Art.  24. 

If  we  draw  any  number  of  half-rays,  OB,  OC,  etc.,  "-OL, 
the  round  angle  will  still  be  the  sum  of  the  consecutive 
angles  AOB,  BOC,  etc., -•- LOA  ;  hence  we  discover  and 
enounce  this 

Theorem  IV.  —  T/w  sum  of  the  consecutive  angles  about  a 
point  in  a  plane  is  a  round  angle. 

N.B.  We  cannot  apply  Axiom  i  immediately,  because 
we  do  not  know,  except  by  Art.  24,  what  is  meant  by  a  sum 
of  angles. 

35.  In  the  foregoing  article  we  have  exemplified  the 
erotetic,  questioning,  investigative  method,  in  which  the  result 


28  GEOMETRY.  [Th.  IV. 

is  not  announced  until  it  is  actually  discovered  and  estab- 
lished. In  Theorems  I.,  II.,  III.,  on  the  other  hand,  the 
dogmatic  procedure  was  illustrated,  the  fact  or  proposition 
being  announced  beforehand,  while  the  demonstration  fol- 
lowed after.  Each  method  has  its  merits,  and  we  shall 
employ  both. 

36.  As  OB  turns  round  from  the  upper  to  the  under 
side  of  OA,  the  angle  A  OB  begins  by  being  less  than  BOA 
and  ends  by  (Fig.  19)  being  greater  than  BOA.     The  plane 


Fig.  19. 

is  continuous,  the  turning  is  continuous,  the  change  m  size 
is  continuous ;  hence,  in  passing  from  the  stage  of  being 
less  to  the  stage  of  being  greater,  the  angle  has  passed 
through  the  intermediate  stage  of  being  equal;  let  OA^  be 
the  position  of  the  rotating  half-ray  at  this  stage  of  equality, 
then  AOA'  =  A'OA.  Two  equal  parts  making  up  a  whole 
are  called  halves;  hence  AOA'  and  A'OA  are  halves  of 
the  full  angle  AOA  ;  they  are  named  straight  (or  flat) 
angles. 

37.    Now,  —  Halves  of  equals  are  equal ; 

All    straight    angles    are    halves    of    equals 
(namely,  equal  round  angles)  ; 


Th.  VII.]  CONGRUENCE.  29 

Hence 

Theorem  V. — All  straight  angles  are  equal. 

This  argument  here  given  in  extenso  is  a  specimen  of  a 
syllogism  (o-vXA.oyto-/>io?  =  computation  =  thinking  together). 
The  first  two  propositions  are  called  premisses,  the  third 
and  last,  in  which  the  other  two  are  thought  together,  is 
called  conclusion.  All  reasoning  may  be  syllogized,  but 
this  is  rarely  done,  as  being  too  formal  and  tedious. 

38.  Theorem  VI. — Two  counter  half- rays  bound  a 
straight  angle. 

A 


O 

Fig.  20. 

For,  let  OA  and  OA^  be  two  such  counter  half-rays  (Fig. 
20)  forming  the  whole  ray  AA\  Turn  the  upper  half  of 
the  plane  film  round  O  as  pivot  until  the  upper  OA^  falls  on 
the  lower  0A\  then,  since  the  ray  is  reversible,  the  ray 
AA^  will  fit  exactly  on  the  ray  A^A ;  i.e.  the  two  angles 
AOA'  and  A'OA  are  congruent  and  equal;  and  the  two 
compose  the  round  angle  A  OA ;  hence  each  is  half  of 
AOA  ;  i.e.  each  is  a  straight  angle,     q.  e.  d. 

39.  Theorem  VII. — Conversely,  The  half-rays  bounding 
a  straight  angle  a7'e  counter. 


P 

Fig.  21. 


Let  OA  and  OA^  bound  a  straight  angle  (Fig.  21)  AOA' ; 
also  let  TB  and  TB'  be  two  counter  half-rays ;    then  they 


30  GEOMETRY.  [Th.  VII. 

bound  a  straight  angle  BPB\  by  Theorem  VL  Since  all 
straight  angles  are  congruent,  we  may  fit  these  two  on  each 
other;  i.e.  we  may  fit  OA  and  OA^  on  PB  and  PB^ ;  but 
BB^  is  a  ray ;  so  then  is  AA^ ;  i.e.  OA  and  OA^  are 
counter,     q.  e.  d. 

40.  We  may  define  a  straight  angle  as  an  angle  bounded  by 
counter  half-rays.     Then  we  may  prove  Theorem  V.  thus  : 

The  ends  of  all  straight  angles  are  pairs  of  counter  half- 
rays  (or  form  whole  rays)  ; 

But  all  such  pairs  (or  whole  rays)  are  congruent  (by 
Theorem  I.) ; 

Therefore,  all  ends  of  straight  angles  are  simultaneously 
congruent. 

But  when  the  ends  of  angles  are  (simultaneously)  congru- 
ent, so  are  the  angles  themselves. 

Hence  all  straight  angles  are  congruent,     q.  e.  d. 

Here  the  first  conclusion,  introduced  by  "  therefore,"  is 
deduced  from  two  premisses  ;  but  the  second,  introduced  by 
"hence,"  is  apparently  deduced  from  only  one.  Only 
apparendy,  however ;  for  one  premiss  was  understood  but 
not  expressed  ;  namely,  all  straight  angles  are  angles  ivhose 
ends  are  congruent.  Without  some  such  implied  additional 
premiss,  it  would  be  impossible  to  draw  the  conclusion. 
Such  a  maimed  syllogism,  with  only  one  expressed  premiss, 
is  called  an  enthymeme.  The  great  body  of  our  reasoning 
is  enthymematic.  We  shall  frequently  call  for  the  suppressed 
premiss  or  reason  by  a  parenthetic  question  (Why?). 

41.  Now  draw  two  rays,  LV  and  MM\  meeting  at  O. 
Each  divides  the  round  angle  about  O  into  two  equal 
straight  angles,  and  together  they  (Fig.  22)  form  four  angles 
a,  /3,  a',  (^'.  Two  angles,  as  a  and  /?,  that  have  a  common 
arm,  are  called  adjacent.     Accordingly  we  see  at  once  : 


Th.  IX.]  CONGRUENCE.  31 

Theorem  VIII.  —  Where  tuio  fays  intersect,  the  sum  of  two 
adjacent  angles  is  a  straight  angle. 


Fig.  22. 

Two  angles  whose  sum  is  a  straight  angle  are  called 
supplemental ;  two  angles  whose  sum  is  a  round  angle  we 
may  call  explemental.  Two  angles  as  a  and  «',  the  arms  of 
the  one  being  counter  to  the  arms  of  the  other,  are  called 
opposite,  or  vertical,  or  counter. 

Theorem  IX.  —  Whoi  two  rays  meet,  the  opposite  angles 
formed  are  equal. 

For  a-{-  (3  =  S  (a  straight  angle)  (why?)  ;  and  «'H-y8  =  S 
(why?). 

Hence  a  -{-  /3  =  a'  -{-  /3  (why?)  ;  therefore  a  =  a'.  Simi- 
larly let  the  student  show  that  jS  =  /3'.     Q.  e.  d. 

An  important  special  case  is  when  the  adjacents,  a  and  /?, 
are  equal.  Each  then  is  half  of  a  straight  angle,  and  there- 
fore one  fourth  of  a  round  angle  ;  and  each  is  called  a  right 
angle.  Now  let  the  student  show  that  if  a  —  /?,  then  «'  =  y8 
and  a  =  /3',  or 

Corollary.  When  two  intersecting  rays  make  two  equal 
adjacent  angles,  they  make  all  four  of  the  angles  equal  (Fig. 
23)- 

Def.  Rays  that  make  right  angles  with  one  another  are 
called  normal  (or  perpendicular)  to  each  other.  N.B. 
The  normal  relation  is  mutual.     How  ? 


32 


GEOMETRY. 


[Th.  IX. 


Def.   Two  angles  whose  sum  is  a  right  angle  are  called 
complemental. 


iy 


Fig.  23. 

42.  Are  we  sure  that  through  any  point  on  a  ray  we  can 
draw  a  normal  to  the  ray?  Let  O  be  any  point  on  the 
ray  LV   (Fig  24).     Let  any  half-ray,  pivoted  at    O,  start 


Fig.  24. 


from  the  position  OL  and  turn  counter-clockwise  into  the 
position  0L\  At  first  the  angle  on  the  right  is  less  than  the 
angle  on  the  left,  at  last  it  is  greater ;  the  plane,  the  turning, 
and  the  angle  are  all  continuous ;  hence  in  passing  from  the 
stage  of  being  less  to  the  stage  of  being  greater,  it  passes 


Th.  X.]  CONGRUENCE.  ZZ 

through  the  stage  of  equaUty.  Let  OR  be  its  position  in 
this  stage  ;  then  ^LOR  =  ^  ROL' ;  i.e.  OR  is  normal  to 
LV.  Moreover,  in  no  other  position,  as  OS^  is  the  ray 
normal  to  LL' ;  for  LOS  is  not  =  LOR  unless  (9»S  falls  on 
ORf  but  is  less  than  L  OR  when  OS  falls  within  the  angle 
LOR,  while  SOL'  is  greater  than  LOR  ;  hence  LOS  and 
SOL'  are  not  equal ;  i.e.  OS  is  not  normal  to  LL'  when  OS 
falls  not  on  OR.  Similarly,  when  L  OS  is  greater  than  L  OR. 
Hence 

Theorem  X.  — Through  a  point  on  a  ray  one,  and  only  one, 
ray  ca7i  be  drawn  normal  to  the  ray. 

43.  Def.  A  ray  through  the  vertex  of  an  angle,  and 
forming  equal  angles  with  the  arms  of  the  angle,  is  called 
the  inner  Bisector  or  mid-ray  of  the  angle.  The  inner 
bisector  of  an  adjacent  supplemental  angle  is  called  the  outer 
bisector  of  the  angle  itself.  Thus  OL  bisects  innerly  and 
OE  bisects  outer ly  the  angle  AOB  (Fig.  25). 


Exercise.    Prove  that  there  is  one  and  only  one  such  inner 
mid-ray. 


34  GEOMETRY.  [Th.  XL 

44.  Theorem  XI.  —  The  inner  Bisector  of  an  angle 
bisects  also  its  explement  innerly. 

Proof.  Let  6>/ bisect  1^  AOB  innerly;  then  '^AOI= 
^  lOB;  call  each  a;  then  a+BOI'=a  +  AOr  (why?)  ; 
take  away  a  ;  then  BOV=AOr  (why?)  ;  i.e.  the  ray  //' 
bisects  innerly  the  angle  BOA,  the  explement  of  A  OB. 
Show  that  the  angles  marked  a'  are  equal. 

45.  Theorem  XII.  —  The  inner  and  outer  Bisectors  of 
an  angle  are  normal  to  each  other. 

Proof.  Let  01  and  OE  bisect  (Fig.  25)  innerly  and 
outerly  the  angle  A  OB.  Then,  by  definition,  the  angles 
marked  a  are  equal,  and  the  angles  marked  ^  are  equal ; 
also  the  sum  of  -{-  a -\-  a  -\-  ^  -\-  fi  =  S ;  hence  a-{-  [i=^S; 
or,  lOE  =  a  right  angle,     q.  e.  d. 

TRIANGLES. 

46.  Thus  far  we  have  treated  only  of  rays  intersecting  in 
a  single  point.     But,  in  general,  three  rays  Z,  M,  N  (Fig. 


Fig.  26, 


26)  will  meet  in  three  points,  since  each  pair  will  meet  in 
one  point,  and  there  are  three  pairs  :  {MN) ,  {NL) ,  {LM) . 


Th.  XIIL] 


TRIANGLES. 


35 


Denote  these  points  by  A,  B,  C.  Then  the  figure  formed 
by  these  three  rays  is  called  a  triangle,  trigon,  or  three-side. 
A,  B,  C  are  its  vertices ;  a,  /3,  y  its  inner  angles ;  BC,  CA, 
AB,  its  inner  sides,  or  simply  its  sides.  Its  angles  and  sides 
are  called  its  parts.  It  is  the  simplest  closed  rectihnear 
figure,  and  most  important.  If  instead  of  taking  three 
rays  we  take  three  points  A,  B,  C,  then  we  may  join  them 
in  pairs  by  rays;  and  since  there  are  three  pairs,  BC,  CA, 
AB,  then  there  are  three  rays,  which  we  may  name  Z,  M,  N. 
Thus  we  see  that  three  points  determine  three  rays,  just  as 
three  rays  determine  three  points.  This  equivalent  deter- 
mination of  the  figure  by  the  same  number  of  points  as  of 
rays  makes  the  figure  unique  and  especially  important.  We 
denote  it  by  the  symbol  A.  We  now  ask,  When  are  two 
triangles  congruent? 

47.  Theorem  XIII.  —  Tim  A  having  two  sides  and  the 
included  angle  of  the  one  equal  respectively  to  two  sides  and 
the  included  angle  of  the  other  are  congruent. 

The  data  are:  Two  A,  ABC  and  A^B^C,  having  the 
three  equalities,  AB  =  A^B\  -AC^A'C,  «  =  cc'  (Fig.  27). 


Fig.  27. 


Proof.    Fit  the  angle  n  on  the  angle  a' ;     this  is  possible, 
because  the  angles  are  equal  and  congruent.     Then  A  falls 


36 


GEOMETRY. 


[Th.  XIV. 


on  ^' ;  also  the  point  B  falls  on  B^  (why?  Because  AB 
=  A'B'),  and  C  falls  on  C  (why?).  Hence  the  three  ver- 
tices of  the  two  A  coincide  in  pairs ;  therefore  the  three 
sides  of  the  two  A  coincide  in  pairs  (why  ?  Because  through 
two  points,  as  A  {A')  and  B  {B'),  only  one  ray  can  pass). 

Q.  E.  D. 

Corollary  i.  The  other  parts  of  the  two  A  are  equal 
or  congruent  in  pairs  of  correspondents  :  ^  =  ;8',  y  =  y', 
BC=B'C\ 

Corollary  2.  Pairs  of  equal  parts  lie  opposite  to  pairs  of 
equal  parts. 

48.  Theorem  XIV.  —  Two  A  having  two  angles  and  the 
included  side  of  the  one  equal  respectively  to  two  angles  and 
the  included  side  of  the  other  are  co figment  (Fig.  28). 


Fig.  28. 


Data :    Two 
AB=:A'B'. 


A  ABC,  A'B'C,   having   «  =  «',   (3  =  (S\ 


Proof.  Fit  AB  on  A' B' ;  this  is  possible  (why?).  Then 
a  will  fit  on  a'  (why?),  and  y8  on  /?'  (why?)  ;  i.e.  the  ray 
^C  will  fit  on  A'C,  and  the  ray  BC  on  B'C.  Then  the 
point  C  will  fall  on  C  (why?  Because  two  rays  meet  in  only 
one  point)  ;     i.e.  the  two  A  fit  exactly,     q.  e.  d. 


Th.  xvi.j 


TRIANGLES, 


37 


49.  We  may  now  use  the  conditiofis  of  congrueitce  thus 
far  estabUshed  to  generate  new  notions  that  may  be  used  in 
estabHshing  other  Theorems. 

Def.  The  ray  normal  to  a  tract  at  its  mid-point  is  called 
the  mid-normal  of  the  tract. 

Theorem  XV.  — Any  point  on  the  mid-normal  of  a  tract  is 
equidistant  from  its  ends  (Fig.  29). 


PaC 


Data :  AB  a  tract,  M  its  mid-point,  L  the  mid-normal, 
F  any  point  on  it. 

Proof.  Compare  the  A  AFAf  and  BFM.  We  have 
AM^BM  (why?).  FM^FM,  ^  AMF='4.  BMF 
(why?)  ;     hence  the  A  are  congruent  (why?)  ;    and  FA  = 

FB.      Q.  E.  D. 

Def.  A  A  with  two  equal  sides,  like  AFB,  is  called 
isosceles ;  the  third  side  is  called  the  base,  and  its  opposite 
angle  the  vertical  angle. 

50.  Theorem  XVI.  —  The  angles  at  the  base  of  an  isosceles 
A  are  equal;  and  conversely. 


38  GEOMETRY.  [Th.  XVI. 

Data:  ABC  an  isosceles  A,  AB  its  base,  AC  and  BC  its 
equal  sides  (Fig.  30). 


Fig.  30. 

Proof.  Take  up  the  A  AB  C,  turn  it  over,  and  replace  it 
in  the  position  BCA.  Then  the  two  A  ACB  and  BCA 
have  the  equal  vertical  angles,  C  and  C,  also  the  side  AC ~ 
BC  (why?)  and  BC  =  AC  (why?)  ;  hence  they  are  con- 
gruent (why  ?) ,  and  the  '^A  =  '^B.     q.  e.  d. 

Conversely,  A  A  whose  basal  angles  are  equal  is  isosceles. 
Let  the  student  conduct  a  proof  quite  similar  to  the  fore- 
going. 

Def.  The  ray  through  a  vertex  and  the  mid-point  of  the 
opposite  side  is  called  the  medial  of  that  side. 

Corollary  i .  In  an  isosceles  A  the  medial  of  the  base  is 
normal  to  it,  and  is  the  mid-ray  of  the  vertical  angle. 

Corollary  2.  When  the  medial  of  a  side  of  a  A  is  normal 
to  the  side,  the  A  is  isosceles.     Prove  it. 

Corollary  3.  When  the  medial  of  a  side  bisects  the 
opposite  angle,  the  A  is  isosceles.     Can  you  prove  it  ? 


Th.  XVI.]  LOGICAL  DIGRESSION.  39 

LOGICAL   DIGRESSION. 

51.  When  the  subject  and  predicate  of  a  proposition  are 
merely  exchanged,  the  proposition  is  said  to  be  converted, 
and  the  new  proposition  is  called  the  cotiverse.  Thus  X  is 
Y;  conversely y  Y\%  X.  In  general,  converses  of  true  prop- 
ositions are  not  true,  but  false.  Thus,  The  horse  is  an 
animal  is  always  correct,  but  The  animal  is  a  horse  is 
generally  false.  A  proposition  remains  true  after  simple 
conversion  only  when  subject  and  predicate  are  properly 
quantified,  thus:  All  horses  are  some  animals;  conversely. 
Some  ani7nals  are  all  horses.  Both  propositions  are  correct 
and  mean  the  same  thing.  But  they  are  awkward  in  ex- 
pression, and  such  forms  are  rarely  or  never  used.  When 
the  quantifying  word  is  all  or  its  equivalent,  the  term  is 
said  to  be  taken  universally ;  when  it  is  some  or  its  equiva- 
lent, the  term  is  said  to  be  taken  particularly.  Thus  in  the 
foregoing  example  horse  is  taken  universally,  but  animal 
particularly.  The  only  useful  conversions  are  of  proposi- 
tions in  which  both  subject  and  predicate  are  universal.  In 
the  great  body  of  propositions  only  the  subject  is  quantified 
universally,  the  quantifier  is  omitted  from  the  predicate,  but 
a  particular  one  is  understood.  To  show  that  a  universal 
quantifier  is  admissible  requires  in  general  a  distinct  proof. 

52.  In  order  to  convert  an  hypothetic  proposition,  we 
exchange  hypothesis  and  conclusion.  Thus,  '\iX  is  F,  Uvs,  V; 
the  converse  is,  if  6^  is  V,  X  is  K  All  such  hypothetic 
propositions  may  be  stated  categorically ^  thus  :  All  cases  of 
X  being  Y  are  cases  of  U  being  V ;  conversely.  All  cases  of 
U  being  V  are  cases  of  X  being  Y.  This  converse  is  plainly 
false  except  when  the  quantifier  all  is  admissible  in  the  first 
predicate. 


40  GEOMETRY.  [Th.  XVII. 

53.  But  while  the  converse  of  a  true  hypothetic  propo- 
sition is  generally  false,  the  contrapositive  is  always  true. 
This  latter  is  formed  by  exchanging  hypothesis  and  conclu- 
sion and  denying  both.  Thus  :  If  Jf  is  F,  then  U  \%  V \ 
contrapositive,  If  U  is  not  V,  then  X  is  not  K  Or,  if  a 
point  is  on  the  mid-normal  of  a  tract,  then  it  is  equidistant 
from  the  ends  of  the  tract ;  contrapositive,  If  a  point  is  not 
equidistant  from  the  ends  of  a  tract,  then  it  is  not  on  the 
mid-normal  of  the  tract. 


54.  Theorem  XVII.  — An  outer  angle  of  a  t^  is  greater 
than  either  inner  non-adjacent  angle. 

Data:  Let  ABC  be  any  A,  a'  an  outer  angle,  /3'  a  non- 
adjacent  inner  one  (Fig.  31). 


Proof.  Draw  the  medial  CM  and  lay  off  MD  =  MC ; 
also  draw  AD.  Then  in  the  A  AMD  and  BMC  we  have 
AM=BM  {why}),  MD  =  MC  (why?),  and  ^AMD  = 
^BMC  (why?)  ;  hence  the  A  are  congruent  (why?),  and 
^MBC^-^  MAD  (why  ?) .  But  ^  MAD  is  only  part  of 
the  ^  «' ;  hence  «'>  ^  MAD  (why?)  ;  i.e.  a'>(3'.     Q.  e.  d. 

Similarly,  prove  that  a'  >  y. 


Th.  xviil]  triangles.  41 

55.  Theorem  XVIII .  — If  t7vo  sides  of  a  A  are  utiequal, 
then  the  opposite  angles  are  unequal  in  the  same  sense  {i.e. 
the  greater  angle  opposite  the  greater  side)  (Fig.  32). 


Fig.  32. 

Data:  ABC  a  A,  AC>AB,  AR  the  mid- ray  of  the 
angle  at  A,  AB'  laid  off  =  AB. 

Proof.  ABR  and  AB^R  are  congruent  (why?)  ;  hence 
^  ABR  =  ^  AB^R  (why  ?)  ;  but  ^  AB^R  >  C  (why  ?)  ; 
i.e.  '^.ABC>^ACB.     Q. e. d. 

Conversely,  If  two  angles  of  a  A  are  unequal,  the  opposite 
sides  are  unequal  in  the  same  sense. 

Proof.  The  opposite  sides  are  not  equal ;  for  when  the 
sides  are  equal,  the  opposite  angles  are  equal  (Theorem 
XVI.),  and  contrapositively,  when  the  angles  are  unequal, 
the  opposite  sides  are  unequal.  Then,  by  the  preceding 
Theorem,  the  greater  angle  Hes  opposite  the  greater  side. 

56.  Join  BB^ ;  then  AR  is  the  mid-normal  of  BB'  (why  ?), 
and  hence  angle  CBB^=z  angle  BB'R  (why  ?) .  Hence  angle 
BB'C>B'BC  (why?);  hence  BC>B'C  (why?).  But 
BC=AC-AB;  htnct  BC>  AC  -  AB ;  i.e. 


42 


GEOMETRY. 


[Th.  XIX. 


Theorem  XIX.  —  Any  side  of  a  A  is  greater  than  the 
difference  of  the  other  two. 

Add  AB  to  both  sides  of  this  inequahty  and  there  results 
AB  +  BC>AC\  i.e. 

Theorem  XX.  —  Any  side  of  a  A  is  less  than  the  sum  of 
the  other  two. 

This  fundamental  Theorem  is  here  proved  on  the  sup- 
position that  AB  <AC;  if  AB  were  =  ^  C  or  >  ^  C,  it  would 
need  no  formal  proof. 

57.  Theorem  XXI.  — A  point  not  on  the  mid-normal  of 
a  tract  is  not  equidistant  from  the  ends  of  the  tract. 

Data :  AB  the  tract,  MN  the  mid-normal,  Q  any  point 
not  o^MN  {Y\g.  33). 


Fig.  33. 

Proof.  Draw  QA  and  QB ;  one  of  them,  as  QA,  must 
cut  MN^X.  some  point,  as/^.  Then  QB<  QP^PB  (why?), 
and  PB  =  PA  (why  ?)  ;  hence  QB  <  QP+  PA  ;  i.e. 
QB  <  QA.     Q.  E.  D. 

Of  what  Theorem  is  this  the  converse  ? 

If  now  we  seek  for  a  point  equidistant  from  A  and  B,  we 
can  find  it  on  the  mid-normal  of  AB  and  only  there  ;  hence 
the  locus  of  a  point  equidistant  from  the  ends  of  a  tract  is 
the  mid-normal  of  the  tract. 


Th.  XXIII.] 


TRIANGLES. 


43 


58.  Theorem  XXII. — Two  A  with  the  three  sides  of  the  one 
equal  respectively  to  the  th7'ee  sides  of  the  other  are  congruent. 

Data:  ABC  and  AB'C  the  two  A,  and  AB  =  A'B\ 
BC=^B^C\    C^=  C'^'  (Fig.  34). 


Fig.  34. 

Proof.  Turn  the  A  A'B'C  over  and  fit  AB^  on  AB  so 
that  O  shall  fall  (say)  at  D.  Draw  CD.  Then  A  and  B 
are  on  the  mid-normal  of  CD  (why?)  ;  hence  the  ray  AB 
is  the  mid-normal  of  CD  (why?)  ;  hence  the  angle  CAB  =1 
angle  DAB,  and  angle  CBA  =  angle  DBA  (why  ?) .  Hence 
the  A  are  congruent  (why?),     q.e.b. 

N.B.  As  to  when  the  A  must  be  turned  round  and  when 
turned  over,  see  Art.  94. 

59.  Theorem  XXIII.  —  A.  J^rom  any  point  outside  of  a 
ray  one  normal  may  be  drawn  to  the  ray. 

Data:    Pthe  point,  LV  the  ray  (Fig.  35). 

Proof.  From  P  draw  a  ray  far  to  the  left,  as  PA,  making 
the  angle  PAL  >  angle  PAD.  Now  let  the  ray  turn  about 
/*  as  a  pivot  into  some  position  far  to  the  right,  making 
angle  PAL  <  PAD.  The  plane,  the  angle,  the  motion,  all 
being  continuous,  in  passing  from  the  stage  of  being  unequal 


GEOMETRY. 


[Th.  XXIII. 


in  one  sense  to  the  stage  of  being  unequal  in  the  opposite 
sense,  the  angles  made  by  the  moving  ray  with  the  fixed  ray 
must  have  passed  through  the  stage  of  equality.     Let  PN  be 


Fig.  35. 

the  ray  in  this  position  so  that  angle  PNL  =  angle  PNV ; 
then  each  is  a  right  angle  by  Definition,  and  /Wis  normal  to 

LL\      Q.  E.  D. 

B.  There  is  only  one  ray  through  a  fixed  point  and  normal 
to  a  fixed  ray. 

Proof.  Any  other  ray  than  PA^,  as  PD,  is  not  normal  to 
LV ;  for  the  outer  angle  PDL  is  >  the  right  angle  PND 
(why  ?) .     Q.  E.  D. 

C.  The  normal  tract  PN  is  shorter  than  any  other  tract 
from  P  to  the  ray  LL' . 

Proof.  For  the  right  angle  at  iV^is  >  angle  PDN  (why  ?)  ; 
hence  PN  <  PD  (why  ?) .     q.  e.  d. 

D.  E.  Equal  tracts  from  point  to  ray  meet  the  ray  at 
equal  distances  from  the  foot  of  the  normal;  and  conversely. 

Proof.  For,  if  DPD^  be  isosceles,  then  the  normal  /Wis 
the  medial  of  the  base  (why?). 

F.  Two,  and  only  two,  tracts  of  given  length  can  be  drawn 
from  a  point  to  a  ray. 


Th.  XXIV.] 


TRIANGLES. 


45 


Proof.  For  two,  and  only  two,  points  are  on  the  ray  at  a 
given  distance  from  the  foot  of  the  normal. 

G.  Of  tracts  drawn  to  points  unequally  distant  from  the 
foot  of  the  normal^  the  one  drawn  to  the  remotest  is  the 
longest. 

Proof.  In  the  A  PDA,  angle  PDA  >  PAD  (why?)  ; 
hence  PA  >  PD  (why  ?) .     q.  e.  d. 

Similarly,  PA'  >  PD. 

H.  Equal  tracts  from  the  point  to  the  ray  make  equal 
angles  with  the  normal  from  the  point  to  the  ray  and  also 
equal  angles  with  the  ray  itself ;  and  conversely. 

I.  Of  unequal  tracts  from  the  point  to  the  ray,  the  longest 
makes  the  greatest  angle  with  the  normal  and  the  least  with 
the  ray. 

Let  the  student  conduct  the  proof  of  H  and  /. 

60.  Theorem  XXIV.  —  Two  A  having  two  angles  and  an 
opposite  side  of  one  equal  respectively  to  two  angles  and  an 
opposite  side  of  the  other  are  congruent. 

Data :  ABC  and  A'B'C  two  A  having  AB  =  A'B\  angle 
a  =  angle  a',  angle  y  =  angle  y'  (Fig.  36). 


Fig.  36. 


46 


GEOMETRY, 


[Th.  XXIV. 


Proof.  Fit  a'  on  «  ;  then  B'  falls  on  B  (why  ?) ,  and  A^  C 
falls  along  A  C.  Draw  the  normal  BN.  Then  B  C  and  B  C 
make  the  same  angle,  y  =  y',  with  the  ray  AN;  hence  they 
are  =  and  meet  the  ray  in  the  same  point  (why  ?)  ;  i.e.  C 
falls  on  C ;   i.e.  the  A  are  congruent.     Q.  e.  d. 

6i.  We  now  come  to  the  so-called  ambiguous  case,  of 
two  A  with  two  sides  and  an  opposite  angle  in  one  equal  to 
the  two  sides  and  the  corresponding  opposite  angle  in  the 


Fig.  37. 

other.    Let  ABC  and  A' B' C  (Fig.  37)  be  the  two  A,  with 
AB  =  A'B\  BC  =  B'C,  and  angle  a  =  angle  a'.     Fit  a'  on 


Th.  XXV.] 


TRIANGLES. 


47 


a ;  then  AB^  falls  on  AB,  B'  on  B ;  but  since  from  a  point 
B  {B')  we  may  draw  two  equal  tracts  to  the  ray  AL,  the 
side  B'  C  may  be  either  of  these  equals  and  may  or  may  not 
fall  on  BC.  In  general,  then,  we  cannot  prove  congruence 
in  this  case.  But  if  ^C  be  >  AB,  then  angle  «  >  angle  y 
(why?),  and  there  is  only  one  tract  on  the  right  of  AB  drawn 
from  B  to  the  ray  AC  and  equal  to  ^C;  the  other  tract 
equal  to  BC  must  be  drawn  outside  of  AB  and  to  the  left. 
Hence  in  this  case,  when  the  angle  lies  opposite  the  greater 
side,  the  A  are  congruent.     Hence 

Theorem  XXV.  —  Two  A  having  two  sides  and  an  angle 
opposite  the  greater  in  one  equal  to  two  sides  and  an  angle 
opposite  the  greater  side  in  the  other  are  congruent. 

Corollary.  Two  right  A  having  a  side  and  any  other 
part  of  one  equal  to  a  side  and  the  corresponding  part  of 
the  other  are  congruent. 


Fig.  38. 

62.    We  have  seen  (Art.  47)  that  when  two  A  have  two 
sides  and  included  angle    in  one  equal  to  two  sides  and 


48  GEOMETRY.  [Th.  XXVI. 

included  angle  in  the  other,  they  are  congruent.  But  what 
if  the  included  angles  are  not  equal  ?  Let  ABC  and  A^B^ C 
be  the  two  A,  having  AB^AB\  BC=  B'C,  but  ji  >  (i'. 
Slip  the  upper  film  of  the  plane  along  until  A^B'  fits  on  AB 
and  let  C  fall  on  D.  Draw  the  mid-ray  BM  of  the  angle 
CBD,  let  it  cut  AC  ^.i  M,  and  draw  DM.  Then  the  A 
CBMd^vA  DBMdiTQ  congruent  (why?)  ;  hence  AM-\-  MB 
=  AC  (why?),  and  AC>AI?,  or  AOA'C.     Hence 

Theorem  XXVI.  —  Two  A  having  two  sides  of  one  equal 
to  two  sides  of  the  other,  but  the  included  angles  unequal, 
have  also  the  third  sides  unequal,  the  greater  side  lying 
opposite  the  greater  angle. 

Conversely,  Two  A  having  two  sides  in  one  equal  to  two 
sides  in  the  other,  but  the  third  sides  unequal,  have  the 
included  angles  also  unequal,  the  greater  angle  being  opposite 
the  greater  side. 

Proof.  The  included  angles  are  not  equal;  for  if  they 
were  equal,  the  A  would  be  congruent  (why?)  and  the 
three  sides  would  be  equal.  Hence  the  included  angles  are 
unequal,  and  the  relation  just  established  holds  ;  namely,  the 
greater  angle  lies  opposite  a  greater  side.     q.  e.  d. 

63.  Theorem  XXVII. — Every  point  on  a  mid-ray  of  an 
angle  is  equidistant  from  its  sides. 

Data :    O  the  angle,  MM'  the  mid-ray,  /*any  point  on  it. 

Proof.  From  P  draw  the  normals  PC  and  PD ;  they  are 
(Fig.  39)  the  distances  of  P  from  the  ends  of  the  angle. 
Then  the  APOC  and  POD  are  congruent  (why?)  ;     hence 

PD  =  PC.      Q.  E.  D. 


Th.  XXVIIL] 


TRIANGLES. 


49 


Conversely,  A  point  equidistant  from  the  ends  of  an  angle 
is  on  a  mid-ray  of  the  angle  (Fig.  39). 


Fig.  39. 


Proof.  If  FC=^FD,  then  the  A  FOC  and  FOB  are 
congruent  (why  ?)  ;  hence  angle  FOD  =  angle  FO  C.   Q.  e.  d. 

Accordingly  we  say  that  the  mid-rays  of  an  angle  are  the 
locus  of  a  point  equidistant  fro  fn  its  ends. 

*64.  It  is  just  at  this  stage  in  the  development  of  the 
doctrine  of  the  Triangle  that  we  are  compelled  to  halt  and 
introduce  a  new  concept  before  we  can  proceed  any  further. 
The  necessity  of  this  step  will  appear  from  what  follows 
(which  may,  however,  be  omitted  on  first  reading,  at  the 
option  of  teacher  or  student) . 

Def  Two  A  not  congruent  are  called  equivalent  when 
they  may  be  cut  up  into  parts  that  are  congruent  in  pairs. 

Theorem  XXVIII. — Any  A  is  equivalent  to  another  A 
having  the  sum  of  two  of  its  angles  equal  to  the  smallest 
angle  of  the  given  A. 

Data :  ABC  the  A,  a  the  least  angle  (Fig.  40). 


50  GEOMETRY,  [Th.  XXIX. 

Proof.  Through  M,  the  mid-point  oi  BC,  draw  AM  a.nd 
make  MD  =  MA.  Then  the  A  A  CM  and  DBM  are  con- 
gruent (why?),  the  part  AMB  is  common  to  ABC  and 
ABD,  and  the  sum  of  the  angles  ADB  and  ^^Z)  =  angle 

BAC.      Q.E.D. 


B 
Fig.  40. 

Corollary.     The  sum  of  the  angles  in  the  new  A  is  equal 
to  the  sum  of  the  angles  in  the  old  A. 

*  65.  We  may  now  repeat  this  process,  applying  it  to  the 
smallest  angle,  as  A,  of  the  A  ABD.  In  the  new  A  ABE 
the  smallest  angle,  as  A,  cannot  be  greater  than  \  of  the 
original  angle  a  in  ABC ;  after  n  repetitions  of  this  process 
we  obtain  a  A,  as  ALB,  in  which  the  sum  of  the  angles  A 

and   L   cannot  be  >  —   of  the  original  angle  a  in  the  A 

2" 

ABC.     By  making  n  as  large  as  we  please,  we  make  — 

as  small   as  we   please,  and   so  we   make   —   of  angle  a 

smaller  than  any  assigned  magnitude  no  matter  how  small. 
Meantime  the  other  angle  B  has  indeed  grown  larger  and 
larger,  but  has  remained  <  a  straight  angle.  Hence  the 
sum  of  the  angles  in  the  A  ALB  cannot  exceed  a  straight 
angle  by  any  amount  however  small;  but  the  sum  of  the 
angles  in  ALB  =  sum  of  the  angles  in  ABC ;  hence 

Theorem  XXIX.  —  77ie  sum  of  the  angles  in  any  A  can- 
not exceed  a  straight  angle  by  any  finite  amount. 


Th.  XXX.] 


TRIANGLES. 


51 


Corollary  i.  The  outer  angle  of  a  A  is  not  less  than  the 
sum  of  the  inner  non-adjacent  angles. 

Corollary  2.  From  any  point  outside  of  a  ray  there  may 
be  drawn  a  ray  making  with  the  given  ray  an  angle  small  at 
will. 

Proof.  From  P  draw  any  ray  PA,  and  lay  off  AB  =  PA 
(Fig.  41).     Then   the   angle    PBA    is    not    greater    than 


Fig.  41. 

\PAN  (why?);  now  lay  off  BC  =  PB  (why?);  then 
angle  PCB  is  not  >  ^  angle  PBA  (why?)  ;  proceeding 
this  way,  we   obtain  after  n  constructions  an  angle  PLN 

not  >  —  of  the  angle  PAN,  and  by  making  n  large  enough 
we  may  make  this  —  as  small  as  we  please,     q.  e.  d. 

2** 

*66.  Theorem  XXX.  —  If  the  sum  of  the  angles  in  any  A 
equals  a  straight  angle,  then  it  equals  a  straight  angle  in 
every  A  (Fig.  42). 


52 


GEOMETRY. 


[Th.  XXX. 


Hypothesis:    ABC  a   A 
A^-B-\-C^S. 


with   the    sum   of  its    angles 


Proof,  (i)  Draw  any  ray  through  C,  as  CD.  Then  if 
the  sum  of  the  angles  in  the  A  ACD  and  BCD  \i^  S—x 
and  S—y^  x  and  y  being  any  definite  magnitudes  however 
small,  then  on  adding  these  sums  we  get  2S—{x-^y); 
and  on  subtracting  the  sum,  S,  of  the  supplemental  angles 
at  Z>  we  get  S  —  {x-\-y)  for  the  sum  of  the  angles  of  the 
A  ABC.  Now  if  this  sum  be  S,  then  x  and  y  must  each 
be  O ;  i.e.  the  sum  of  the  angles  in  each  of  the  A  A  CD 
and  BCD  is  S.  Now  draw  DE  and  DF\  in  each  of  the  four 
small  A  the  sum  of  the  angles  is  still  =6".  (2)  We  may 
now  make  a  A  as  large  as  we  please  and  of  any  shape  what- 
ever, but  the  sum  of  the  angles  will  remain  =  S.  For,  take 
the  same  A  ABC,  and  draw  CD  normal  to  AB.  Then  the 
sum  of  the   (Fig.  43)  angles  in  the  A  ACD  is  ^S",  as  has 


Fig.  43. 

been  shown  above  ;  also  angle  Z>  is  a  right  angle  ;  hence 
the  angles  A  and  ACD  are  complementary.  Now  along 
A  C  fit  another  A  A  CD^  congruent  with  A  CD ;  then  all 
the  angles  of  the  quadrilateral  ADCD^  are  right,  and  the 
figure  is  called  a  rectangle.  Now  we  can  place  horizontally 
side  by  side  as  many  of  these  rectangles,  all  congruent,  as  we 
please,  say  /  of  them ;  we  can  also  place  as  many  of  them 
vertically,  one  upon  another,  as  we  please,  say  q  of  them ; 


Th.  XXX.]  TRIANGLES.  53 

and  we  can  then  fill  up  the  whole  figure  into  a  new  rectan- 
gle, as  large  as  we  please.  About  each  inner  junction-point 
of  the  sides  of  the  rectangles  there  will  be  four  right  angles 
plainly.  Now  connect  the  two  opposite  vertices,  as  A  and  Z, 
of  this  rectangle.  So  we  get  two  congruent  right  A,  in  each 
of  which  the  sum  of  the  angles  is  S.  Then  any  A  that  we 
cut  off  from  this  right  A  will,  by  the  foregoing,  have  the 
sum  of  its  angles  equal  to  S.  Since  p  and  q  are  entirely  in 
our  power,  we  may  make  in  this  way  any  desired  right  A 
and  from  it  cut  off  any  desired  oblique  A,  with  the  sum  of 
its  angles  =  S.     q.  e.  d. 

Hence  either  no  A  has  the  sum  of  its  angles  =  S,  or 
every  A  has  the  sum  of  its  angles  =  S. 

67.  A  logical  choice  between  these  alternatives  is  impos- 
sible, but  the  matter  may  be  cleared  up  by  the  following 
considerations  : 

Across  any  ray  ZM  draw  a  transversal  T,  cutting  LM  at 
O,  and  making  the  angles  a,  jS,  y,  8.  Through  any  point, 
as  O'j  of  7" draw  a  ray  (Fig.  44)  L'M'  making  angle  «'  =  a. 


Fig.  44.  - 

This  is  evidently  possible  (why  ?) .      Then  plainly  /8'  =  ^, 
y'  =  y,  8'  =  8,  a'  =  a ;  they  are  called  corresponding  angles  ; 


5+  GEOMETRY.  [Th.  XXX. 

also  a  and  y',  /?  and  8'  are  equal,  —  they  are  called  alternate 
angles ;  also  a  and  8',  as  well  as  /8  and  y',  are  supplemental, 
—  they  are  called  interadjacent  angles. 

68.  Now  let  /*be  the  mid-point  of  00^ ;  on  it  as  a  pivot 
turn  the  whole  right  side  of  the  plane  round  through  a 
straight  angle  until  O  falls  on  (9',  and  (9'  falls  on  O.  Then, 
since  the  angles  about  O  and  O^  are  equal  as  above  stated, 
the  half-ray  OL  will  fall  and  fit  on  the  half-ray  O^M\  and 
the  half-ray  O^V  on  the  halt-ray  OM.  Accordingly,  if  the 
rays  LM  and  VM^  meet  on  one  side  of  the  transversal  T, 
they  also  meet  on  the  other  side  of  T. 

69.  Three  possibilities  here  lie  open  : 

(i)  The  rays  ZJ/and  Z'J/'  may  meet  on  the  left  and 
also  on  the  right  of  T,  in  different  points. 

(2)  They  may  meet  on  the  left  and  ajso  on  the  right  of 
T,  in  the  same  point. 

(3)  They  may  not  meet  at  all. 

No  logical  choice  among  these  three  is  possible.  But 
in  all  regions  accessible  to  our  experience  the  rays  neither 
converge  nor  show  any  tendency  to  converge.  Hence  we 
assume  as  an 

Axiom  A.  Two  rays  that  make  with  any  third  ray  a  pair 
of  corresponding  angles  equal,  or  a  pair  of  alternate  angles 
equal,  or  a  pair  of  interadjacent  angles  supplemental,  are 
non-inter sectors. 

70.  But  another  query  now  arises.  Is  it  possible  to  draw 
another  ray  through  6>'  so  close  to  V  that  it  will  not  meet 
OL  however  far  both  may  be  produced?  Here  again  it  is 
impossible  to  answer  from  pure  logic.  An  appeal  to  experi- 
ence is  all  that  is  left  us.     This  latter  testifies  that  no  ray 


Th.  XXX.]  PARALLELS.  55 

can  be  drawn  through  (9'  so  close  to  (9'Z'  as  not  to  approach 
and  finally  meet  the  ray  OL.     Hence  we  assume  as  another 

Axiom  B.  Through  any  point  in  a  plane  only  one  non- 
intersector  can  be  drawn  for  a  given  straight  line. 

This  single  non-intersector  is  commonly  called  the  parallel, 
through  the  point,  to  the  straight  line. 

71.  It  cannot  be  too  firmly  insisted,  nor  too  distinctly 
understood,  that  the  existence  of  any  non-intersector  at  all, 
and  the  existence  of  only  one  for  any  given  point  and  given 
ray,  are  both  assumptions,  which  cannot  be  proved  to  be 
facts.  The  best  that  can  be  said  of  them,  and  that  is  quite 
good  enough,  is  that  they  and  all  their  logical  consequences 
accord  completely  and  perfectly  with  all  our  experience  as 
far  as  our  experience  has  hitherto  gone.  Even  then,  if 
there  be  any  error  in  our  assumptions,  we  have  thus  far  been 
utterly  unable  to  find  it  out. 

A  geometry  that  should  reject  either  or  both  of  these 
assumptions  would  have  just  as  much  logical  right  to  be  as 
the  geometry  that  accepts  them,  and  such  geometries  lack 
neither  interest  nor  importance.  They  may  be  called  Hyper- 
Euclidean  in  contradistinction  from  this  of  ours,  which  from 
this  point  on  is  Euclidean  (so-called  from  the  Greek  master, 
Euclides,  who  distinctly  enunciated  the  equivalent  of  our 
Axioms  in  a  Definition  and  a  Postulate). 

Note.  —  Observe  the  relation  of  Axioms  A  and  B :  the  one  is  the 
converse  of  the  other. 

Observe  also  that  the  necessity  of  assuming  the  first  lies  in  our  igno- 
rance of  the  indefinitely  great,  and  the  occasion  of  assuming  the  other 
lies  in  our  ignorance  of  the  indefinitely  small.     See  Note,  Art.  301. 

72.  Accepting  our  Axioms  as  at  least  exacter  than  any 
experiment  we  can  inake,  we  may  now  easily  settle  the  ques- 


56  GEOMETRY.  [Th.  XXXI. 

tion  as  to  the  sum  of  the  angles  in  a  A.  Let  ABC  be  any 
A ;  through  the  vertex  C  draw  the  one  parallel  to  the  base 
AB.  Then  a=a\  (3  =  ft'  (why?)  ;  also  a'  +  y -^  ft' =  S ; 
hence  a-\-y  +  ft  =  S;  i.e.  (Fig.  45) 

Theorem  XXXI.  —  The  sum  of  the  angles  in  a  A.  is  a 
straight  angle. 


Fig.  45. 

Corollary  i.  The  outer  angle  E  equals  the  sum  of  the 
inner  non-adjacent  angles  a  and  y  (why?). 

Corollary  2.  If  two  angles  of  a  A  be  known,  the  third  is 
also  known. 

Corollary  3.  If  two  A  have  two  angles,  or  the  sum  of  two 
angles  of  the  one  equal  to  two  angles,  or  the  sum  of  two 
angles  of  the  other,  then  the  third  angles  are  equal. 

Corollary  4.  To  know  the  three  angles  of  a  A  is  not  to 
know  the  A  completely,  for  many  A  may  have  the  same 
three  angles.  Such  A  are  similar,  as  we  shall  see,  but  are 
not  congruent ;  they  are  alike  in  shape,  but  not  in  size. 

73.  Next  to  normahty,  parallehsm  is  the  most  important 
relation  in  which  rays  can  stand  to  each  other,  and  we  must 
now  use  the  new  relation  in  the  generation  of  new  concepts. 


Th.  XXXIII.] 


PARALLELOGRAMS. 


57 


Theorem  XXXII. — Parallel  Intercepts  between  parallels 
are  equal. 

Data :   L  and  L\  J/ and  M ,  two  pairs  of  parallels  (Fig.  46). 


Fig.  46. 

Proof.  Draw  BD.  Then  the  A  ABD  and  CDB  are 
congruent  (why?),  and  AB  =  CD,  BC=  DA.     q.  e. d. 

Def.  The  figure  ABCD  formed  by  two  pairs  of  parallel 
sides  is  called  a  parallelogram,  and  may  be  denoted  by  the 
symbol  O. 

A  join  of  opposite  vertices,  as  BD,  is  called  a  diagonal. 

74.    Theorem  XXXIII.  —  Properties  of  the  parallelogram. 

A.  The  opposite  sides  of  a  parallelogram  are  equal. 
This  has  just  been  proved. 

B.  The  opposite  angles  of  a  parallelogram  are  equal. 
Proof.     «  =  y8  (why?);   p=a'  (why?);   hence  «  =  «'. 

Q.  E.  D. 

Corollary.  Adjacent  angles  of  a  parallelogram  are  sup- 
plementary. 

C.  Each  diagonal  of  a  parallelogram  cuts  it  into  two  con- 
gruent A.     Prove  it. 


58 


GEOMETRY. 


[Th.  XXXIV. 


D.    The  diagonals  of  a  parallelogram  bisect  each  other 
(Fig.  47)' 


FIG.  47. 

Proof.     The  A  AMB  and  CMD  are  congruent  (why?)  ; 
hence  AM^  CM,  BM=  DM.     q.  e.  d. 

75.    We  may  now  convert  all  the  foregoing  propositions 
and  obtain  as  many  criteria  of  the  parallelogram. 

Theorem  XXXIV.  —  A'.   A  4-side  with  its  opposite  sides 
equal  is  a  parallelogram. 

Data  :    AB  =  CD,  AD=^  CB  (Fig  48). 


Fig.  48. 

Proof.  Draw  BD.  Then  ABD  and  CDB  are  congruent 
(why?);  hence  ;8  =  8 ;  and  AD  and  CB  are  parallel; 
similarly,  AB  and  CD  are  parallel;  hence  ABCD  is  a  par- 
allelogram.    Q.  E.  D. 


Th.  XXXIV.]  PARALLELOGRAMS.  59 

B'.   A  4-side  with  opposite  angles  equal  is  a  parallelogram. 
Data:    (^  =  a',  y8  + /S' =  y  +  y'  (Fig.  49)- 


^^-^i 


Fig.  49. 

Proof.  Since  a  =  «',  /?  +  y  =  ^'  +  y'  (why  ?) .  Hence 
^  =  y'j  ;S'  =  y ;  z.<f.  opposite  sides  are  parallel,  the  4-side 
is  a  parallelogram,     q.  e.  d. 

(Z\  A  4-side  that  is  cut  by  each  diagonal  into  two  congru- 
ent A  is  a  parallelogram. 

For  the  opposite  angles  must  be  equal  (why?)  ;  hence, 
etc.      Q.  E.  D. 

D'.  A  4-side  whose  diagonals  bisect  each  other  is  a 
parallelogra  m . 

For  the  opposite  sides  are  equal,  being  opposite  equal 
angles  in  congruent  A  ;  hence,  etc.     q.  e.  d. 

E'.  A  4-side  with  one  pair  of  sides  equal  and  parallel  is 
a  parallelogram. 

For  the  other  two  sides  are  equal  and  parallel  (why?)  ; 
hence,  etc.     q.  e.  d. 

76.  The  foregoing  properties  and  criteria  of  the  parallel- 
ogram illustrate  excellently  the  nature  of  a  definition.     This 


60  GEOMETRY.  [Th.  XXXV. 

latter  defines  or  bounds  off  by  stating  something  that  is  true 
of  the  thing  defined,  but  of  nothing  else.  Accordingly,  the 
characteristic  of  every  definition  or  definitive  property  is 
that  the  proposition  that  states  it  may  be  converted  simply. 
Thus  : 

Every  parallelogram  is  a  4-side  with  opposite  angles 
equal;  and  conversely,  every  4-side  with  opposite  angles 
equal  is  a  parallelogram. 

Not  every  property  is  definitive,  and  hence  not  every 
property  may  be  used  as  test  or  criterion. 

77.    Special  Parallelograms. 

Def.   An  equilateral  parallelogram  is  called  a  rhombus. 

Theorem  XXXV.  —  The  diagonals  of  a  rhombus  are  nor- 
mal to  each  other. 

Let  the  student  conduct  the  proof  suggested  by  the  figure 
(Fig- 50)- 


Fig.  50. 

Conversely,  A  parallelogram  ivhose  diagonals  are  normal 
to  each  other  is  equilateral^  or  a  rhombus.  Let  the  student 
supply  the  proof. 

78.  Def.  An  equiangular  parallelogram  is  called  a  rect- 
angle (for  all  the  angles  are  right  angles) . 


Th.  XXXVII. ]  PARALLELOGRAMS. 


61 


Theorem  XXXVI.  —  The  diagonals  of  a  rectangle  are  equal 
(Fig.  51). 


D 

C 

A 

B 

Fig.  51. 
For  the  A  ABC  and  BAD  are  congruent  (why?)  ;  hence 

AC=BD.      Q.E.D. 

Conversely,  A  parallelogram  with  equal  diagonals  is  equi- 
angular, or  a  rectangle. 

For  the  A  ABC  and  BAD  are  again  congruent,  though 
for  another  reason.     What  reason  ? 

79.  Def.  A  parallelogram  both  equilateral  and  equi- 
angular is  called  a  square. 

Theorem  XXXVII.  —  The  diagonals  of  a  square  are  equal 
and  normal  to  each  other. 


Fig.  52. 


62  GEOMETRY.  [Th.  XXXVIII. 

For  the  square,  being  both  rhombus  and  rectangle,  has 
all  the  definitive  properties  of  both.  Or  the  student  may 
prove  the  proposition  directly  from  the  figure  (Fig.  52),  as 
well  as  its  converse  : 

A  parallelogram  with  diagonals  equal  and  normal  to  each 
other  is  a  square. 

80.  Can  we  convert  Theorem  XXXII.  and  prove  that 
equal  intercepts  between  parallels  are  parallel  ?  Manifestly 
no  (Fig.  53),  for  from  the  point  C  we  may  draw  two.  equal 


Fig.  53. 

tracts  to  the  other  parallel,  the  one  CB  parallel  to  AD,  the 
other  CB^  sloped  at  the  same  angle  to  the  parallels  but  in 
opposite  ways.  We  may  call  CB^  anti-parallel  to  AD,  and 
the  figure  AB^CD  an  anti-parallelogram.  Since  from  any 
point  C  only  two  equal  tracts,  or  tracts  of  given  length,  may 
be  drawn  to  the  other  parallel  through  A,  we  have  the 

Theorem  XXXVIII.  —  Equal  intercepts  between  parallels 
are  either  parallel  or  anti-parallel. 

Corollary  i.  Adjacent  angles  of  an  anti-parallelogram 
are  alternately  equal  or  supplemental. 

Corollary  2.  Anti-parallels  prolonged  meet  at  the  vertex 
of  an  isosceles  A. 


Th.  XL.] 


GENERAL    QUADRILATERAL. 


63 


THE   GENERAL   QUADRILATERAL   OR   4-SIDE. 

81.  A  Quadrilateral  is  determined  by  four  intersecting 
rays.  These  determine  six  points,  the  four  inner  vertices, 
C,  Drf  £f  F,  and  the  two  outer  ones,  A^  B.  The  cross-rays, 
CE^  DF,  AB,  are  the  diagonals,  CE  and  DF  inner,  AB 
outer.  Commonly  the  outer  diagonal  is  little  used,  and  the 
inner  ones  are  called  the  diagonals.  When  none  of  the 
angles  C,  D,  E,  F,  of  the  4-side  is  greater  than  a  straight 
angle,  the  4-side  is  called  the  normal,  as  CDEF.  It  is  the 
only  form  ordinarily  considered.  The  other  two  forms  are 
(2)  the  crossed,  ACBE,  and  (3)  the  inverse,  AD  BE 
(Fig.  54).     For  all  forms  let  the  student  prove 


Fig.  54. 

Theorem  XXXIX.  —  The  sum  of  the  inner  angles  of  a 
^-side  is  a  round  angle. 

Corollary.     When   two  angles   of  a   4-side  are   supple- 
mental, so  are  the  other  two. 


82.    Theorem  XL.  —  The  angles  between  two  rays  equal 
the  angles  between  two  normals  to  the  7-ays. 


64 


GEOMETRY. 


[Th.  XLI. 


Data :    OL  and  OM  any  two  rays,  PA  and  PB  any  two 
normals  to  them  (Fig.  55). 


/ 

/ 

X 

p 

/ 

B 

/ 

M 

Fig.  55. 

Proof.  The  angles  at  A  and  B  are  right  angles  and 
therefore  supplemental  (why?)  ;    hence  a  —  a',  and  /?  ==  /3'. 

Q.  E.  D. 

N.B.  The  4-side  with  its  opposite  angles  supplemental  is 
very  important  and  has  received  the  name  encyclic  4-side, 
for  reasons  to  be  seen  later  on  (Arts.  126-7). 


THREE   OR   MORE   PARALLELS. 

83.  Theorem  XLI.  —  Th7'ee  parallels  that  make  equal 
intercepts  on  one  transversal^  fnake  equal  intercepts  on  any 
transversal. 

Data  :  Z,  M,  N,  three  parallels,  and  AB  =  BC,  and  DBF 
any  transversal  (Fig.  56). 

Proof.  Draw  D^EF'  parallel  to  ABC.  Then  AB=^BC 
(why?),  AB  =  D'E  (why?),  and  BC^EP  (why?); 
hence  D^E  =  EF  (why?),  hence  the  A  DED'  and  FEF 
are  congruent  (why?) ;    hence  DE  =  EF  (why?),     q.  e.  d. 


Th.  xlii.]     three  or  more  parallels.  65 

84.  Def.  A  4-side  formed  by  two  parallels  and  two 
transversals  is  called  a  trapezoid.  Thus  A  CFD  is  a  trape- 
zoid. The  parallel  sides  are  called  the  bases  (major  and 
minor)  ;  the  parallel  through  the  mid-points  of  the  trans- 
verse sides  is  the  mid-parallel. 


Fig.  56. 

Theorem  XLII.  —  The  mid-parallel  of  a  trapezoid  equals 
the  half-sum  of  its  bases. 

Let  the  student  elicit  the  proof  from  the  foregoing  figure. 

Corollary  i.  A  parallel  to  a  base  of  a  A  bisecting  one 
side  bisects  also  the  other.     {Hint.     Let  D  fall  on  A.) 

Corollary  2.  A  ray  bisecting  two  sides  of  a  A  is  parallel 
to  the  third. 

For  only  one  ray  can  bisect  two  sides  (why?),  and  we 
have  just  seen  (Cor.  i)  that  a  ray  parallel  to  the  base  does 
this  ;  hence,     q.  e.  d. 

Corollary  3.  The  mid-parallel  to  the  base  of  a  A  equals 
half  the  base. 

85.  Def.  Three  or  more  rays  that  pass  through  a  point 
are  said  to  concur  or  be  concurrent. 


66  GEOMETRY,  [Th.  XLIII. 

Theorem  XLIII.  —  The  me  dials  of  a  A  concur. 

Data :    ABC  a  A,  AFand  BQ  two  medials  (Fig.  57). 


Fig.  57. 

Proof.  Draw  a  ray  from  C  through  O,  the  intersection 
of  the  two  medials,  and  lay  off  OH=  CO.  Draw  AH  and 
BH\  they  are  parallel  to  BQ  and  AP  (why?)  ;  hence 
A  OBH  is  a  parallelogram  (why  ?)  ;  hence  AR  =  BR 
(why?).  Hence  COR  is  the  third  medial;  2>.  the  three 
medials  pass  through  O.     q.  e.  d. 

Corollary.  Each  medial  cuts  off  a  third  from  each  of 
the  other  two.     For  C0  =  20R  (why?). 

Def.  The  point  of  concurrence  of  the  medials  is  called 
the  centroid  of  the  A.  It  is  two-thirds  the  length  of  each 
medial  from  the  corresponding  vertex. 

86.  Theorem  XLIV.  —  The  mid-normals  of  the  sides  of 
a  A  concur. 

Data  :  ABC  a  A,  Z  and  J/ mid-normals  to  the  sides  BC 
and  CA,  meeting  at  S. 


Th.  XLV.] 


CONCURRENTS. 


67 


Proof.  6*  is  equidistant  from  B  and  C  (why?),  and  from 
Cand  A  (why?)  ;  hence  S  is  equidistant  from  A  and  B 
(why?),  or  is  on  the  mid-normal  of  AB  (why?)  ;  hence 
the  mid-normals  concur  (Fig.  58).     Q.  e.  d. 


Fig.  58. 

Corollary.  S  is  equidistant  from  A,  B,  and  C,  and  no 
other  point  in  the  plane  is  (why?). 

Def.  The  point  of  concurrence  of  the  mid-normals  is 
called  the  circumcentre  of  the  A. 

87.  Def.  A  tract  from  a  vertex  of  a  A  normal  to  the 
opposite  side  is  called  an  altitude  of  the  A.  Sometimes, 
when  length  is  not  considered,  the  whole  ray  is  called  the 
altitude. 

Theorem  XLV.  —  The  altitudes  of  a  A  concur. 

Proof.  Using  the  preceding  figure,  draw  the  A  A^B^C. 
Its  sides  are  parallel  to  the  sides  of  ABC  (why?)  ;  hence 
its  altitudes  are  the  mid-normals  Z,  J/,  N;  and  these  have 
just  been  found  to  concur.  Also,  since  ABC  may  be  any 
A,  A'B'C  may  be  any  A ;  hence  the  altitudes  of  any  A 
concur,     q.  e.  d. 


68  GEOMETRY.  [Th.  XLVI. 

Def.  The  point  of  concurrence  of  altitudes  is  called  the 
orthocentre  (or  alticentre)  of  the  A. 

Def.  In  a  right  A  the  side  opposite  the  right  angle  is 
called  the  hypotenuse  (  =  subtense  =  under-stretch) . 

Queries :  Where  do  circumcentre  and  orthocentre  lie : 
(i)  in  an  acute-angled  A  ?  (2)  in  an  obtuse-angled  A  ? 
(3)  in  a  right  A  ? 

88.  Theorem  XLVI.  — The  inner  mid-rays  of  the  angles 
of  a  A  concur. 

Data :  ABC  a  A,  AL,  BM,  CN  the  inner  mid-rays  of  its 
angles  (Fig.  59). 


Fig.  59. 

Proof.  Let  AL  and  BM  intersect  at  /.  Then  /  is  equi- 
distant from  AB  and  AC,  and  from  AB  and  BC  (why?)  ; 
hence  /  is  equidistant  from  AC  and  BC \  hence  /  is  on 
the  inner  mid-ray  of  the  angle  C ;  i.e.  the  three  inner  mid- 
rays  concur  in  /.     q.  e.  d. 

Def  The  point  of  concurrence  of  the  inner  mid-rays  is 
called  the  in-centre  of  the  A. 


Th.  XLVIL]  exercises  L  69 

89.    Theorem  XLVII.  —  The  outer  mid-rays  of  two  angles 
and  the  inner  mid-ray  of  the  other  angle  of  a  A  concur. 
Let  the  student  conduct  the  proof  (Fig.  60). 


Fig.  60^ 


Def.  The  points  of  concurrence  are  called  ex-centres  of 
the  A  :  there  are  three. 

EXERCISES   I. 

Little  by  little  the  student  has  been  left  to  rely  more  and 
more  upon  his  own  resources  of  knowledge  and  ratiocination 
in  the  conduct  of  the  foregoing  investigations.  He  has  now 
possessed  himself  of  a  large  fund  of  concepts,  and  he  must 
test  his  ability  to  wield,  combine,  and  manipulate  them  in 
forging  original  proofs  of  theorems.  Let  him  bear  always 
in  mind  the  fundamental  logical  principle  that  every  example 


70  GEOMETRY. 

of  a  general  concept  has  all  the  marks  of  thai  general  concept. 
Let  him  begin  his  proof  by  stating  precisely  the  data,  the 
given  or  known  facts,  let  him  draw  a  corresponding  diagram 
in  order  to  have  a  clearer  view  of  the  spatial  relations  in- 
volved, let  him  note  carefully  what  concepts  are  present  in 
the  proposition,  let  him  draw  auxiliary  lines  and  introduce 
auxiliary  concepts  at  pleasure.  But  let  him  exhaust  simple 
means  before  trying  more  complicated,  let  him  distinguish, 
by  manner  of  drawing,  the  principal  from  the  auxiliary  rays, 
and  especially  let  him  be  systematic  and  consistent  in  the 
literation  of  his  figures. 

1.  How  many  degrees  in  a  straight  angle?  In  a  right 
angle  ? 

Historical  Note.  —  For  purposes  of  computation  the  round  angle 
is  divided  into  360  equal  parts  called  degrees,  each  degree  into  60 
equal  minutes  (partes  minutcB  primse),  each  minute  into  60  equal 
seconds  (partes  minutae  secundce),  denoted  by  °,  ',  "  respectively. 
This  sexagesimal  division  is  cumbrous  and  unscientific,  but  is  apparently 
permanently  established.  It  seems  to  have  originated  with  the  Baby- 
lonians, who  fixed  approximately  the  length  of  the  year  at  360  days,  in 
which  time  the  sun  completed  his  circuit  of  the  heavens.  A  degree, 
then,  as  is  indicated  by  the  name,  which  means  s:tep  in  Latin,  Greek, 
Hebrew  {gradus,  ^ad/xos  (or  T/Mrjfxa),  ma'a/ak),wa.s  primarily  the  daily 
s^ep  of  the  sun  eastward  among  the  stars.  The  Chinese,  on  the  other 
hand,  determined  the  year  much  more  exactly  at  365^  days,  and 
accordingly,  in  defiance  of  all  arithmetic  sense,  divided  the  circle  into 
365^  degrees. 

2.  The  angles  of  a  A  are  equal ;  how  many  degrees  in 
each? 

Remark.  —  Such  a  A  is  called  equiangular,  more  commonly  equi- 
lateral, but  better  still  regular. 

3.  Show  that  this  regular  A  is  equilateral. 

4.  One  angle  of  a  A  is  a  right-angle ;  the  others  are 
equal ;  how  many  degrees  in  each  ? 


EXERCISES  I.  71 

5.  One  angle  of  a  A  is  twice  and  the  other  thrice  the 
third  ;  what  are  the  angles  ? 

6.  Two  angles  of  a  A  are  measured  and  found  to  be 
46°  37' 24"  and  52° 48' 39";  what  is  the  third? 

7.  One  angle  of  a  A  is  measured  to  be  61°  22' 40";  the 
others  are  computed  to  be  49°  34'  28"  and  69°  2'  43"  ;  what 
do  you  infer? 

8.  A  half-ray  turns  through  two  round  angles  counter- 
clockwise, then  through  half  a  right-angle  clockwise,  then 
through  a  straight  angle  counter-clockwise,  then  through  \ 
of  a  round  angle  counter-clockwise,  then  through  ^  of  a 
straight  angle  clockwise  ;  what  angle  does  it  make  in  its 
final  position  with  its  original  position  ? 

9.  6^  is  a  fixed  point  (called  origin)  on  a  ray,  A  and  B 
are  any  pair  of  points,  M  their  mid-point.  Show  and  state 
in  words  that  2  0M=  OA  +  OB. 

10.  A,  B,  C  are  three  points  on  a  ray,  A\  B\  C  are 
mid-points  of  the  tracts  BC,  CA,  AB,  and  O  is  any  point 
on  the  ray ;  show  that  0A-{- 0B-\- OC=OA^+  OB^  +  OC. 

11.  A,  B,  C,  D,  O  are  points  on  a  ray;  A\  B\  C  are 
mid-points  of  AB,  BC,  CD)  A",  B",  are  mid-points  of 
A'B',  B'C ;  J/ is  the  mid-point  of  A"B" ;  prove  SOM= 
OA  +  sOB-\-sOC+OZ>. 

12.  What  are  the  conditions  of  congruence  in  isosceles 
A  ?     In  right  A  ? 

13.  In  what  A  does  one  angle  equal  the  sum  of  the 
other  two? 

Def.  A  number  of  tracts  joining  consecutively  any  number 
of  points  (first  with  second,  second  with  third,  etc.)  is  called 
a  broken  line,  or  train  of  tracts,  or  polygon.     Where  the  last 


72  GEOMETRY. 

point  falls  on  the  first  the  polygon  is  closed;  otherwise  it  is 
open.  Unless  otherwise  stated,  the  polygon  is  supposed  to 
be  closed.  The  points  are  the  vertices,  the  tracts  are  the 
sides  of  the  polygon.  The  closed  polygon  has  the  same 
number  of  vertices  and  sides,  and  we  may  call  it  an  n-angle 
or  n-side.  The  angles  between  the  pairs  of  consecutive 
sides  are  the  angles  of  the  polygon,  either  inner  or  outer ; 
unless  otherwise  stated,  inner  angles  are  referred  to.  Inner 
and  outer  angles  at  any  vertex  are  supplemental.  When 
each  inner  angle  is  less  than  a  straight  angle,  the  polygon 
is  called  convex ;  otherwise,  re-entrant.  Unless  otherwise 
stated,  convex  polygons  are  meant.  Sides  and  angles  of  a 
polygon  may  be  reckoned  either  clockwise  or  counter-clock- 
wise. 

14.  Prove  that  the  sum  of  the  inner  angles  of  an  ;«-side 
is  {n  —  2)  straight  angles.  What  is  the  sum  of  the  outer 
angles  ? 

15.  Find  the  angle  in  a  regular  {i.e.  equiangular  and 
equilateral)  3-side,  4-side,  5-side,  8-side,  12-side.  (For 
proof  that  there  is  a  regular  «-side,  see  Art.  137.) 

16.  Show  that  a  (convex)  polygon  cannot  have  more 
than  three  obtuse  outer  angles,  nor  more  than  three  acute 
inner  angles. 

1 7.  Two  angles  of  a  A  are  a  and  ^ ;  find  the  angles  at 
the  intersection  of  their  mid-rays. 

18.  If  two  A  have  their  sides  parallel  or  perpendicular  in 
pairs,  then  the  A  are  mutually  equiangular. 

19.  The  medial  to  the  hypotenuse  of  a  right  A  cuts  the 
A  into  two  isosceles  A. 

20.  An  angle  in  a  A  is  obtuse,  right,  or  acute,  according 
as  the  medial  to  the  opposite  side  is  less  than,  equal  to,  or 
greater  than,  half  the  opposite  side. 


EXERCISES  I.  73 

21.  A  medial  will  be  greater  than,  equal  to,  or  less  than, 
half  the  side  it  bisects,  according  as  the  opposite  angle  is 
acute,  right,  or  obtuse. 

22.  HP  and  Q  be  on  the  mid-normal  of  AB^  then 
AAFQ=i/SBFQ  {=  indicates  congruence). 

23.  AB  is  the  base,  C  the  opposite  vertex  of  an  isosce- 
les A;  show  that  ABN=BAM  (i)  when  AM  and  BN 
are  altitudes,  (2)  when  they  are  medials,  (3)  when  they  are 
mid-rays  of  angles  A  and  B,  (4)  when  MN  is  normal  to  the 
mid-normal  of  AB. 

24.  F  is  any  point  within  the  A  ABC ;  show  that 
AF+BF<AC-^  CB,  AF+FB+  CF>  i  {AB+BC-\-  CA) . 

25.  ABC"' L  and  AB^C-"L  are  two  convex  polygons, 
not  crossing  each  other,  between  the  same  pairs  of  points, 
A  and  L  ;  which  is  the  longer  ?     Give  proof. 

26.  /*  is  a  point  within  AABC ;  show  that  angle  AFB 
>  ACB  and  sum  of  angles  sit  F=  2{A  +  B -\-  C). 

27.  F  is  equidistant  from  A,  B,  and  C;  show  that  angle 
AFB  =2 {Single  ACB). 

28.  Conversely,  if  angle  ^P^  =  2  (angle  ACB),  angle 
BFC=  2  (angle  BAC),  and  angle  CFA  =  2  (angle  CBA), 
then  F  is  equidistant  from  A,  B,  C. 

29.  The  mid-rays  of  the  angles  at  the  ends  of  the  trans- 
verse axis  of  a  kite  cut  the  sides  in  the  vertices  of  an 
anti-parallelogram  (Art.  99). 

30.  The  four  joins  of  the  consecutive  mid-points  of  the 
sides  of  a  4-side  form  a  parallelogram. 

31.  The  joins  of  the  mid-points  of  the  pairs  of  opposite 
sides  and  of  the  pairs  of  diagonals  of  a  4-side  concur,  bisect- 
ing each  other. 


74  GEOMETRY. 

32.  The  mid-parallels  to  the  sides  of  a  A  cut  it  into  4 
congruent  A. 

33.  What  figures  are  formed  by  the  mid-parallels  when 
the  A  is  right?  isosceles?  regular? 

34.  A  parallelogram  is  a  rhombus  if  a  diagonal  bisects 
one  of  its  angles. 

35.  A  parallelogram  is  a  square  if  its  diagonals  are  equal 
and  one  bisects  an  angle  of  the  parallelogram. 

36.  From  any  point  in  the  base  of  an  isosceles  A  parallels 
are  drawn  to  the  sides ;  the  parallelogram  so  formed  has  a 
constant  perimeter  ( =  measure  round  =  sum  of  sides) . 

37.  The  sum  of  the  distances  of  any  point  on  the  base  of 
an  isosceles  A  from  the  sides  is  constant. 

38.  The  sum  of  the  distances  of  any  point  within  a  regular 
A  from  the  sides  is  constant. — What  if  the  point  be  without 
the  A? 

39.  jP  is  on  a  mid-ray  of  the  angle  A  in  the  A  ABC', 
compare  the  difference  of  FB  and  PC :  when  P  is  within 
the  A,  and  when  P  is  without. 

40.  The  inner  mid-ray  of  one  angle  of  a  A  and  the  outer 
mid- ray  of  another  form  an  angle  that  is  half  the  third  angle 
of  the  A. 

41.  6>  is  the  orthocentre  of  the  A  ABC;  express  the 
angles  A  OB,  BOC,  CO  A,  through  the  angles  A,  B,  C. 

42.  Do  the  like  for  the  circum-centre  6* and  the  in-centre  /. 

43.  The  medial  to  the  hypotenuse  of  a  right  A  equals 
one-half  of  that  hypotenuse. 

44.  The  mid-rays  of  two  adjacent  angles  of  a  parallelogram 
are  normal  to  each  other. 


EXERCISES  I.  IS 

45.  In  a  5 -pointed  star  the  sum  of  the  angles  at  the 
points  is  a  straight  angle.  What  is  the  sum  in  a  7 -pointed 
star? 

46.  Parallels  are  drawn  to  the  sides  of  a  regular  A,  tri- 
secting the  sides  ;  what  figures  result  ? 

47.  A  side  of  a  A  is  cut  into  8  equal  parts,  through  each 
section  point  parallels  are  drawn  to  the  other  sides  ;  how  are 
the  other  sides  cut  and  what  figures  result? 

48.  Two  A  are  congruent  when  they  have  two  mid-tracts 
of  two  corresponding  angles  equal,  and  besides  have  equal 

(i)  these  angles  and  a  pair  of  the  including  sides  ;  or 

(2)  two  pairs  of  corresponding  angles  ;  or 

(3)  one  pair  of  corresponding  angles  and  the  correspond- 
ing angles  of  the  mid-tract  with  the  opposite  side  ;  or 

(4)  one  pair  of  including  sides  and  the  adjacent  segment 
of  the  opposite  side. 

49.  Two  A  are  congruent  when  they  have  two  corre- 
sponding sides  and  their  medials  equal,  and  besides  have 
equal 

(i)  another  pair  of  sides  ;  or 

(2)  the  angles  of  the  medial  with  its  side  (in  pairs)  ;  or 

(3)  a  pair  of  angles  of  the  bisected  side  with  another 
side,  the  angles  of  the  medial  with  this  side  being  both 
acute  or  both  obtuse  ;  or 

(4)  a  pair  of  angles  of  the  medial  with  an  including  side, 
the  corresponding  angles  of  the  medial  with  its  side  being 
both  acute  or  both  obtuse. 

50.  Two  A  are  congruent  when  they  have  a  pair  of  cor- 
responding altitudes  equal,  and  besides  have  equal 


76  GEOMETRY. 

(i)   the  pair  of  bases  and  a  pair  of  adjacent  angles ;  or 

(2)  the  pair  of  bases  and  another  pair  of  sides  ;  or 

(3)  the  pairs  of  angles  of  the  altitude  with  the  sides  ;  or 

(4)  two  pairs  of  corresponding  angles  ;  or 

(5)  the  two  pairs  of  sides,  when  the  altitudes  lie  both 
between  or  both  not  between  the  sides  of  the  A. 

SYMMETRY. 

90.  We  have  seen  that  congruent  figures  are  alike  in  size 
and  shape,  different  only  in  place,  and  may  be  made  to  fit 
point  for  point,  line  for  line,  angle  for  angle.  The  parts 
that  fit  one  on  the  other  are  said  to  correspond  or  be  corre- 
spondent. Plainly  only  hke  can  correspond  to  like,  as  point 
to  point,  etc. 

Def.  The  ray  through  two  points  we  may  call  the  join  of 
those  points,  and  the  point  on  two  rays  the  join  of  the  rays. 

91.  It  is  now  plain  that  if  ^  corresponds  to  A  and  B  to 
B\  then  the  join  of  A  and  B  must  correspond  to  the  join 
of  A  and  B^ ;  for  in  fitting  A  on  A'  and  B  on  B'  the  ray 
AB  must  fit  on  the  ray  AB^  (why?).  Also  if  the  ray  Z 
corresponds  to  L\  and  M  to  M',  then  the  join  of  L  and  M 
must  correspond  to  the  join  of  Z'  and  M'  (why?).  These 
facts  are  very  simple  but  very  important. 

We  shall  think  of  the  plane  as  a  thin  double  film,  the  one 
figure  drawn  in  the  upper  layer,  the  other  in  the  lower. 

92.  Two  congruent  figures  may  be  placed  anywhere  and 
any  way  in  the  plane,  but  there  are  two  positions  especially 
important :  ( i )  the  one  in  which  the  one  figure  may  be 
superimposed  on  the  other  by  turning  the  one  half  of  the 
plane  through  a  straight  angle  about  a  ray  called  an  axis ; 
(2)  the  one  in  which  the  one  figure  may  be  fitted  on  the 


SYMMETRY. 


77 


other  by  turning  the  one  half  of  the  plane  through  a  straight 
angle  about  a  point  called  a  centre. 

Congruent  figures  in  either  of  these  two  positions  are 
called  symmetric :  in  the  first  case  axally,  as  to  the  axis  of 
symmetry;  in  the  second  case  centrally,  as  to  the  centre 
of  symmetry. 

93.  In  two  symmetries,  corresponding  angles,  like  all 
other  correspondents,  are  of  course  congruent ;  but  they 
are  reckoned  oppositely  if  the  symmetry  be  axal,  similarly  if 


Fig.  61. 


78  GEOMETRY. 

it  be  central.  To  parallels  correspond  parallels  ;  to  normals, 
normals  ;  to  mid-points,  mid-points  ;  to  mid-rays,  mid-rays  ; 
to  the  axis  corresponds  the  axis,  each  point  to  itself;  to  the 
centre  corresponds  the  centre  itself  (Fig.  6i). 

Elements,  whether  points  or  lines,  that  correspond  to 
themselves  may  be  called  self-correspondent  or  double. 

It  is  also  manifest  that  centre  and  axis  are  the  only  self- 
correspondents  ;  hence  if  a  point  be  self-correspondent,  it 
must  lie  on  the  axis  in  axal  symmetry,  or  be  the  centre  in 
central  symmetry ;  and  if  two  counter  half-rays  be  corre- 
spondent, they  (or  the  ray)  must  be  normal  to  the  axis  in 
axal  symmetry,  or  go  through  the  centre  in  central  sym- 
metry. 

94.  These  facts  are  all  perfectly  obvious,  but  a  more 
vivid  exemplification  of  the  nature  of  these  two  kinds  of 
symmetry  may  perhaps  be  found  in  the  following : 

Suppose  the  axis  of  symmetry  to  be  a  perfect  plane 
mirror;  then  either  half  of  the  plane  may  be  treated  as  the 
reflection  or  exact  image  of  the  other,  and  will  be  the  sym- 
metric of  the  other  as  to  the  mirror-axis.  For  the  image  of 
any  point  A  is  the  point  A  such  that  the  axis  is  the  mid- 
normal  of  AA\  as  we  know  from  Physics ;  also,  on  folding 
over  the  one  half  of  the  plane  about  the  axis  upon  the  other 
half,  the  point  A  falls  on  A^  (why?)  ;  hence  A^  is  the  sym- 
metric of  A  as  to  the  axis. 

Suppose  the  centre  of  symmetry  6"  to  be  also  a  reflector ; 
then  the  reflection  or  image  of  any  point  A  will  be  a  point 
A  such  that  S  is  the  mid-point  of  the  tract  AA\  and  on 
rotation  through  a  straight  angle  about  ^  the  point  A  falls 
on  A\  and  the  half-ray  SA  fits  on  the  half- ray  SA\  Hence 
either  of  two  centrally  symmetric  figures  is  the  exact  image 
of  the  other  reflected  from  the  centre  of  symmetry  S. 


SYMMETRY. 


79 


Note  carefully  that  these  two  species  of  symmetry  depend 
upon  the  two  fundamental  definitive  properties  of  the  plane  : 
central  symmetry  upon  the  homoeoidality  of  the  plane,  axal 
symmetry  upon  the  reversibility  of  the  plane.  Moreover, 
axally  symmetric  figures  can  not  be  fitted  on  each  other 
without  reversion,  folding  over ;  by  movement  in  the  plane 
their  corresponding  parts  can  at  best  be  <?/posed,  but  never 
super^o^^A ;  while  on  the  other  hand  central  symmetries 
may  be  superposed,  but  cannot  be  ^/posed,  along  any  ray, 
by  motion  in  the  plane.  In  central  symmetries  the  corre- 
sponding parts  follow  one  another  in  the  same  order,  but  in 
axal  symmetries  they  follow  in  opposite  orders. 

95.  We  must  now  discuss  these  two  symmetries  more 
minutely,  and  to  exhibit  a  certain  remarkable  relation  hold- 
ing between  them  we  arrange  their  properties  in  parallel 
columns. 


In  Axal  Symmetry. 

1.  The  axis  corresponds  to  it- 
self. 

2.  Every  point  of  the  axis  cor- 
responds to  itself. 

3.  Every  self-correspondent 
point  lies  on  the  axis. 

4.  The  join  of  two  correspond- 
ent rays  is  on  the  axis. 

(For  it  is  self-correspondent.) 

5.  Correspondent  points  are 
equidistant  from  every  point  on 
the  axis. 


In  Central  Symmetry. 

1.  The  centre  corresponds  to 
itself. 

2.  Every  ray  through  the  cen- 
tre corresponds  to  itself  (each  half 
to  the  other). 

3.  Every  self-correspondent  ray 
goes  through  the  centre. 

4.  The  join  of  two  correspond- 
ent points  goes  through  the  cen- 
tre. 

(For  it  is  self-correspondent.) 

5.  Correspondent  rays  are 
equally  inclined  (isoclinal)  to 
every  ray  through  the  centre; 
hence  they  are  parallel,  as  is 
otherwise  manifest. 


80 


GEOMETRY. 


6.  The  axis  is  a  mid-ray  of 
every  angle  between  correspond- 
ent rays,  and  in  fact  the  inner 
mid-ray. 

N.B.  The  outer  mid-ray  is  a 
normal  to  the  axis. 

7.  The  join  of  two  correspond- 
ent/<?m/^  is  a  normal  to  the  axis. 

8.  Correspondent  tracts  are 
anti-parallel. 

9.  Correspondent  points  are 
equidistant  from  the  axis. 

10.  The  join  of  two  rays  and 
the  join  of  their  correspondents 
themselves  correspond. 


6.  The  centre  is  a  mid-point  of 
every  tract  between  correspond- 
ent points,  and  in  fact  the  inner 
mid-point. 

N.B.  The  outer  mid-point  is  a 
point  at  infinity. 

7.  The  join  of  two  correspond- 
ent rays  is  at  infinity. 

(For  they  are  parallel.) 

8.  Correspondent  angles  are 
contra-posed  {i.e.  have  their  arms 
extended  oppositely). 

9.  Correspondent  rays  are 
equidistant  from  the  centre. 

10.  The  join  of  two  points  and 
the  join  of  their  correspondents 
themselves  correspond. 

96.  On  regarding  closely  these  correlated  propositions,  it 
becomes  clear  that  the  one  set  differs  from  the  other  only 
in  the  interchange  of  certain  notions,  as  poi7it  and  ray,  tract 
and  angle,  etc.  Every  property  of  axal  symmetry  has  its 
obverse  in  central  symmetry,  and  vice  versa.  This  most 
profound,  important,  and  interesting  fact  has  received  the 
name  of  the  Principle  of  Reciprocity.  We  make  this  notion 
more  precise  by  the  following 

Def.  Two  figures  such  that  to  every  point  of  each  corre- 
sponds a  ray  of  the  other,  and  to  every  ray  of  each  a  point 
of  the  other,  are  ca//ed  reciprocal.     For  example  : 

Suppose  rays  drawn  through  a  point  O  to  any  number  of 
points,  A,  B,  C,  D,  E,  .  .  .  on  a  ray  L.  Then  the  point 
O  with  its  ray  through  it,  and  the  ray  L  with  its  point  on 
it,  are  two  reciprocal  figures  (Fig.  62).  The  first  is  called 
a  (flat)  pencil  of  rays,  O  being  the  centre ;  the  second  is 
called  a  row  (or  range)  of  points,  L  being  the  axis.     Sup- 


SYMMETRY. 


81 


pose  we  have  now  a  second  pencil  through  (9'  and  a  second 
row  on  Z'.  These  two  figures  are  again  reciprocal,  and  the 
two  pairs  of  reciprocals  together  make  up  another  more 
complex  pair  of  reciprocals.  In  this  latter  pair  we  find  our 
definition  fully  exemplified.  To  O  and  O^  correspond  L  and 
Z' ;  to  the  rays  through  O  and  (9'  correspond  the  points  on  Z 
and  Z' ;  also,  to  the  join  (ray)  of  O  and  O^  corresponds  the 


Fig.  62. 

join  (point)  of  Z  and  Z' ;  to  any  point  as  P,  the  join  of  two 
rays  {OA,  O'A'),  corresponds  a  ray  AA',  the  join  of  two 
points  (A,  A').  So  Q,  R,  S,  Tare  points  corresponding  to 
the  rays  BB',  CC,  DD\  EE\  We  may  notice  further 
that  angle  and  tract  correspond  in  the  reciprocal  figures ; 
thus  the  angle  AOB  corresponds  to  the  tract  AB,  and  the 
angle  BOC  to  the  tract  BC;  while  the  angle  OFO'  corre- 
sponds to  the  tract  A  A'  and  the  tract  RS  to  the  angle 
between  the  rays  corresponding  to  R  and  6" ;  namely,  between 
CC  and  W.  Let  the  student  trace  out  as  many  corre- 
spondences as  possible. 


82  GEOMETRY. 

97.  To  three  points  fixing  a  triangle  in  either  of  two 
reciprocals  must  correspond  also  three  rays  fixing  a  triangle 
in  the  other  reciprocal;  hence,  in  general,  triangle  corre- 
sponds to  triangle  in  reciprocals.  But  notice  :  the  sides  of 
one  correspond  to  the  vertices  of  the  other ;  hence  if  the 
sides  of  one  all  go  through  the  same  point,  the  vertices  of 
the  other  all  lie  on  the  same  ray ;  that  is,  three  concurrent 
rays  in  either  reciprocal  correspond  to  three  coUinear  points 
in  the  other. 

It  now  appears  that  axal  and  central  symmetry  are  recip- 
rocal to  each  other ;  the  reciprocal  of  an  axal  symmetric  is 
a  central  symmetric,  and  the  reciprocal  of  a  central  sym- 
metric is  an  axal  symmetric ;  the  reciprocal  properties  of 
axal  symmetry  are  the  properties  of  central  symmetry,  and 
the  reciprocal  properties  of  central  symmetry  are  the  proper- 
ties of  axal  symmetry. 

Very  often  the  two  symmetric  figures  may  be  regarded 
as  the  two  halves  of  one  figure  ;  this  one  figure  is  then  said 
to  be  symmetric  as  to  the  axis  of  symmetry  or  as  to  the 
centre  of  symmetry,  as  the  case  may  be. 

98.  If  our  figure  be  two  points,  A  and  A\  then  the  mid- 
normal  X  of  the  tract  AA^  is  the  axis  of  symmetry,  mani- 
festly. If,  now,  any  double  point  D  on  the  axis  be  joined 
with  A  and  A\  there  results  the  isosceles  A  ADA\  whence 
it  appears  that  (Fig.  d^i) 

The  isosceles  A  is  a  symmetric  A. 

It  is  plain  that  any  two  points  on  the  ray  AA^  equidistant 
from  N  are  symmetric  as  to  X,  that  all  points  on  the  ray, 
and  indeed  in  the  whole  plane,  may  be  arranged  in  sym- 
metric pairs,  the  members  of  each  pair  equidistant  from  the 
axis  X. 


SYMMETRY. 


83 


99.  Now  take  two  points  on  the  axis,  as  D  and  D\  or 
D  and  Z>",  and  consider  the  4-side  DAD'A\  It  is  com- 
posed of  two  A,  ADD^  and  ADD\  symmetric  with  each 
other  as  to  the  axis  X,  and  opposed  along  that  axis.  Hence 
the  4-side  is  itself  symmetrical  as  to  X 

Def.  Such  a  4-side,  with  an  axis  of  symmetry,  is  called  a 
kite. 

If  we  hold  D  fast,  and  let  Z)'  glide  along  X,  the  4-side 
ADA^D^  remains  a  kite.     We  see  that  there  are  two  kinds 


of  kites,  the  convex  kite,  as  ADA^D\  and  the  re-entrant,  as 
ADAD^\  As  the  gliding  point  passes  through  iV^  the  kite 
changes  from  one  kind  to  the  other,  passing  through  the 
intermediate  form  of  the  symmetrical  A. 

When  the  gliding  point  reaches  a  position  Z>'  such  that 
ND  =  ND\  then  the  four  sides  of  the  kite  are  all  equal 
(why?),  and  the  kite  becomes  a  rhombus  (why?).  In  this 
case  D  and  D^  are  symmetric  as  to  AA^  as  an  axis  of  sym- 


84 


GEOMETRY. 


metry.      Hence  the  rhombus   has  two  axes  of  symmetry ; 
namely,  its  two  diagonals. 

In  all  cases  the  diagonals,  A  A  and  DD\  of  the  kite  are 
normal  to  each  other  (why?). 

100.    Now  consider  a  pair  of  points,  B  and  B\  symmetric 
as  to  the  axis  X  (Fig.  64).     Then  Xis  mid-normal  oi BB\ 


Fig.  64. 

If  C  and  C  be  any  other  pair  of  symmetric  points,  then  X 
is  also  mid-normal  of  CC  ;  hence  BB^  and  CC  are  parallel 
(why?).  Also  the  tracts  BC  and  B^ C  are  symmetric  as 
to  X  (why?),  and  the  4-side  BB^CC  is  itself  symmetric  as 
to  the  axis  X,     Hence  the  angles  at  C  and  C  are  equal. 


SYMMETRY. 


85 


also  the  angles  at  B  and  B^  are  equal  (why?)  ;  hence  the 
angles  at  B  and  C  and  at  B^  and  C  are  supplemental 
(why?),  and  the  4-side  BB'C'C  is  an  anti-parallelogram 
(why  ?) .  Hence  we  see  that  another  symmetric  ^-side  is  an 
anti-parallelogram. 

It  is  plain  that  every  anti- parallelogram  is  symmetric,  for 
we  know  that  the  obHque  sides  prolonged  yield  an  isosceles  A. 
Let  the  student  complete  the  proof. 

loi.  There  is  only  one  kind  of  symmetric  A,  the  isosceles. 
For,  let^^^'  (Fig.  65)  be  symmetric  and  A'  correspondent 


/\ 


Fig.  65. 

to  A.  Then  B  must  correspond  to  itself  (why?)  ;  hence 
B  must  lie  on  the  axis  (why?)  ;  hence  BA  =  BA^  (why?). 

Now  let  the  student  prove  that 

(i)  In  a  symmetric  A  the  axis  of  symmetry  is  a  medial ; 
(2)   it  is  also  a  7nid-ray ;   (3)   it  is  also  a  mid-normal. 


86  GEOMETRY. 

Conversely^  let  him  show  that 

A  medial  that  is  a  mid-ray,  or  a  mid-normal,  is  an  axis 
of  symmetry. 

102.  There  are  only  two  axally  symmetric  4-sides  ;  namely, 
the  kite  and  the  anti- parallelogram.  For,  in  a  symmetric 
4-side  a  vertex  must  correspond  to  a  vertex  (why?).  Also, 
not  all  vertices  can  be  on  the  axis  (why?).  Also,  a  vertex 
on  the  axis  is  a  double  point  (why?).  Also,  the  vertices 
not  on  the  axis  must  appear  in  pairs  (why?)  ;  hence  there 
must  be  either  two  or  four  of  them.  If  there  be  two  only, 
then  the  other  two  are  on  the  axis  and  the  4-side  is  a  kite ; 
if  there  be  four  of  them,  we  have  just  seen  that  the  4-side  is 
an  anti-parallelogram. 

103.  Now  let  us  turn  to  the  reciprocals.  The  reciprocals 
of  the  two  points  A  and  A^  symmetric  as  to  the  axis  X  will 
be  two  rays  L,  L\  symmetric  as  to  the  centre  S.  But  rays 
symmetric  as  to  a  centre  are  parallel  (why?)  ;  hence  we  have 
two  parallels  symmetric  as  to  S,  which  is  midway  between 
them.  The  rays  are  symmetric  as  to  any  other  point  S'  mid- 
way between  them  (why?).  The  piece  of  plane  between 
these  parallels  is  called  a  parallel  strip,  or  band  (Fig.  66). 


^'-^ 

M 

•s' 

L 

Fig.  66. 


SYMMETRY.  87 

But  what  corresponds  to  the  point  D  on  the  axis  X}  The 
answer  is  :  a  ray  R  through  6"  (why?).  Hence  to  the  sym- 
metric A  of  the  three  points  A,  A\  D,  there  corresponds 
the  figure  formed  by  two  parallels  L,  L\  and  a  transverse  R 
through  5,  —  a  so-called  half-strip.  This  is  truly  a  three- 
side,  but  not  apparently  a  A  (3 -angle),  for  the  parallels  do 
not  meet  in  finity,  in  regions  accessible  to  our  experience. 
Hence,  instead  of  saying  that  the  reciprocal  of  a  A  in  axal 
symmetry  is  a  A  (3-angle  or  3-point)  in  central  sym- 
metry, we  should  have  said,  accurately,  that  the  recipro- 
cal of  a  A  in  axal  symmetry  is  a  3-side  (or  trilateral)  in 
central  symmetry,  which  will  always  be  a  A  except  when 
sides  are  parallel  or  all  concur.  In  higher  Geometry  it  is 
very  convenient  to  remove  this  apparent  exception  by  using 
this  form  of  expression  :  the  parallels  meet  not  in  finity,  but 
in  infinity. 

104.  It  is  indeed  plain  that 

A  A  can  have  no  centre  of  symmetry. 

For,  since  vertex  corresponds  to  vertex,  and  since  corre- 
spondents appear  in  pairs,  one  vertex  must  be  a  double  point ; 
hence  it  would  have  to  be  the  centre  S  (why  ?) .  But  the 
other  two  vertices  would  have  to  lie  on  a  ray  through  S,  being 
correspondents  ;  hence  the  three  vertices  would  be  collinear, 
and  the  A  would  be  flattened  out  to  a  triply-laid  ray. 

105.  But  there  is  a  centrally  symmetrical  4-side  ;  namely, 
the  parallelogram.  For,  consider  once  more  the  kite 
AXA^X  and  let  us  reciprocate  it  into  a  centrally  symmetric 
figure  (Fig.  67).  To  the  axis  XX^  will  correspond  the  cen- 
tre S;  to  the  symmetric  pair  of  rays  ^Jfand  y4'X  will  corre- 
spond a  symmetric  pair  of  points  P  and  P ;  to  the  join  of 
those  on  the  axis  X  will  correspond  the  join  of  these  through 


A'^ 


88  GEOMETRY, 

the  centre  {FF^y.  Similarly,  to  the  symmetric  rays  AX'  and 
A'X'  will  correspond  the  symmetric  points  Q  and  Q\  and 
to  the  join  X'  will  correspond  the  join  QQ.  Also,  AX 
and  AX'  have  a  join  A  while  A'X  and  ^'X'  have  a  join  ^', 
and  these  joins  are  symmetric  as  to  the  axis  XX' ;  recipro- 


FlG.  67. 

cally,  /'and  Q  have  a  join  Z*^,  and  F'  and  ^'  have  a  join 
FQ'y  and  these  joins  are  symmetric  as  to  ^S;  that  is,  they  are 
parallel  (why?).  Similarly,  FQ'  and  F' Q  correspond  to  B 
{AX,  A'X')  and  B'  (A'X,  AX')  ;  but  B  and  B'  are  sym- 
metric as  to XX'  (why?)  ;  hence  FQ'  and  F'Qave  symmetric 
as  to  S,  i.e.  are  parallel.  Hence  FQ  FQ  is  symmetric  as 
to  S,  and  is  2,  parallelograin.     q.  e.  d. 


106.    We  may  indeed  see  at  once  that  since  any  two  par- 
allels are  centrally  symmetrical  as  to  any  mid-point,  a  pair 


SYMMETRY. 


89 


of  parallels  or  a  parallelogram  is  symmetric  as  to  the  common 
mid-way  point,  the  intersection  of  the  diagonals.  But  the 
foregoing  reciprocation  is  instructive,  as  illustrating  in  detail 
the  method  to  be  pursued,  and  as  showing  the  intimate  rela- 
tion of  the  different  symmetric  quadrilaterals ;  namely,  the 
parallelogram  is  the  common  reciprocal  of  doth '^tQ  and  axiti- 
parallelogram,  which  are  thus  seen  to  be  really  one. 

107.  Central  symmetry  does  not  in  general  imply  any- 
thing at  all  with  respect  to  axal  symmetry  in  a  figure.  We 
may  draw  through  any  point  S  any  number  of  rays  and  lay 
off  on  each  from  -S  a  pair  of  counter  tracts  SF  and  SP\ 
SQ  and  SQ',  etc.  No  matter  how  FQ,  etc.,  be  chosen,  the 
figure  so  obtained  will  be  centrally  symmetric  as  to  6";  but 
it  may  have  no  axal  symmetry  whatever.  Neither  does  axal 
symmetry  in  general  imply  any  central  symmetry,  but  we 
may  estabhsh  the  following  important 

Theorem.  —  Any  figure  with  two  rectangular  axes  of 
symmetry  has  also  a  centre  of  symr?ietty ;  namely,  the  inter- 
section of  those  axes. 


p 

V 

# 

p 

'^^ 

^.--^ 

^^ 

^^ 

\ 

3'' 

P 

Y 

Fig.  68. 


90  GEOMETRY. 

Data :  XX^  and  FF'  two  rectangular  axes,  P  any  point 
of  a  figure  symmetric  as  to  these  axes  (Fig.  68). 

Proof.  The  point  P^  symmetric  with  P  as  to  XX^  is  a 
point  of  the  figure  (why?)  ;  also  P"  symmetric  with  P^  as 
to  yy  is  a  point  of  the  figure  (why?)  ;  so  too  is  /^'" 
(why?)  ;  the  figure  pp'P^p^'^  is  a  rectangle  (why?),  its 
diagonals  halve  each  other,  and  SP=  SP"  =  SP' =  SP'". 
Hence  ^  is  a  centre  of  symmetry,     q.  e.  d. 

THE   CIRCLE. 

1 08.  We  have  already  discovered  the  existence  of  a 
homoeoidal  plane  curve  not  reversible  and  have  named  it 
circle. 

Defs.  A  ray  cutting  a  curve  is  called  a  secant,  as  L  ;  the 
part  of  the  secant  intercepted  by  the  curve,  or  the  tract 
between  two  points  of  the  curve,  is  called  a  chord,  as  AB. 
A  finite  part  of  a  curve  is  called  an  arc.  A  chord  and  an 
arc  with  the  same  two  ends  are  said  to  subtend  each  other. 
Also,  the  intercept  of  any  line  between  the  ends  of  an  angle 
is  said  to  sub  fern/ the  angle.  Thus  ^Cand  Z>£  subtend  the 
angle  O  (Fig.  69). 


Fig.  69. 


Th.  XLIX.] 


THE    CIRCLE. 


91 


109.  Theorem  XLVIII.  —  Congruent  arcs  subtend  con- 
gruent chords. 

Proof.  Let  the  arcs  AB  and  CD  be  congruent ;  then  we 
may  fit  ^  on  C  and  at  the  same  time  B  on  D ;  then  the 
chords  AB  and  CB  fit  throughout  (why  ?) .     q.  e.  d. 

N.B.  We  can  not  convert  this  proposition  at  once  (why?) 
(Fig,  70). 


Fig.  70. 

no.    Theorem  XLIX.  — A  closed  curve  is  cut  by  a  ray  in 
an  even  number  of  points   (Fig.  71). 


Proof.  Let  Z  be  a  ray,  C  any  closed  curve.  Suppose  a 
point  P  to  trace  out  the  ray  Z.  At  first  P  is  without  the 
curve,  at  last  it  is  also  without  the  curve ;  hence  P  has 
crossed  the  curve  going  out  as  often  as  it  has  crossed  the 
curve  going  in,  for  every  entrance  there  is  an  exit ;  hence 
the  points  of  intersection  appear  in  pairs,  their  number  is 
even,  as  o,  2,  4,  6,  ...  2  n,     q.  e.  d. 


92 


GEOMETRY. 


[Th.  L. 


These  preliminary  or  auxiliary  theorems,  which  prepare 
the  way  for  a  theorem  to  follow,  are  sometimes  called 
lemmas  (XyjfjifjLa  =  assumption,  premise,  support,  prop) . 

*iii.  Theorem  L. — A  circle  has  an  axis  of  symmetry 
through  every  one  of  its  points  (Fig.   72). 


Fig.  72. 

Proof.  Let  D  be  any  point  of  a  circle.  Take  any  arc 
DP,  and  slip  it  round  till  P  falls  on  D  and  D  on  P' ;  this 
is  possible  (why  ?) .  Then  PDP'  is  a  symmetrical  A  (why  ?)  ; 
and  its  axis  of  symmetry  DR  halves  normally  the  chord 
PP\  and  also  halves  the  angle  PDP'  (why?).  Now  take 
any  other  arc  DQ  and  slip  it  round  till  Q  falls  on  D  and  D 
on  Q\  so  that  P)Q  and  QD  are  congruent.  Then  the 
chords  DQ  and  DQ  are  congruent  (why?).  Also,  on 
taking  away  the  congruents  DP  and  DP^  we  have  left  PQ 
and  P^Q  as  congruent  remainders.     Hence  the  chords  PQ 


Th.  LI.]  THE    CIRCLE.  93 

and  PQ  are  congruent  (why?).  Hence  the  A  PDQ  and 
PDQ  are  congruent  (why?)  ;  hence  the  angles  PDQ  and 
PDQ  are  equal  (why?)  ;  hence  DR  halves  also  the  angle 
QDQ  (why?).  But  the  A  QDQ  is  symmetric  (why?)  ; 
hence  DR  is  also  its  axi?  of  symmetry,  and  Q  and  Q  are 
symmetric  points  of  the  circle ;  hence  any  point  of  the 
circle  has  its  symmetric  point  as  to  DR;  i.e.  DR  is  an 
axis  of  symmetry  of  the  circle.  Moreover,  D  was  any  point 
of  the  circle  ;  hence  through  any  point  of  the  circle  passes 
an  axis  of  symmetry,     q.  e.  d. 

Def.  A  ray  halving  a  system  of  parallel  chords  is  called  a 
diameter;  the  chords  and  diameter  are  called  conjugate  to 
each  other. 

Corollary  i.  In  a  circle  a  diameter  is  normal  to  its  con- 
jugate chords. 

Corollary  2.  Every  mid-normal  to  a  chord  in  a  circle  is 
a  diameter  and  halves  the  subtended  arcs. 

*ii2.  Theorem  LI.  — A  circle  has  a  centre  of  symmetry 
(Fig.  72). 

For  the  ray  through  D  must  cut  the  circle  in  some  second 
point,  as  R  (why?),  and  as  the  ray  turns  round  from  the 
position  DR  to  the  reversed  position  RD,  through  a  straight 
angle,  it  must  pass  through  some  position,  QQ,  normal  to 
its  original  position  (why  ?) .  Hence  for  any  axis  of  symmetry 
there  is  another  normal  thereto  and  their  intersection  is  a 
centre  of  symmetry  (why  ?) .     q.  e.  d. 

N.B.   There  is  only  one  centre  of  symmetry  (why?). 

Def.  This  centre  of  symmetry  is  named  centre  of  the 
circle.  It  is  often  convenient  to  call  the  whole  ray  through 
the  centre  a  centre  ray  or  line^  and  to  restrict  the  term 
diameter  to  the  centre  chord. 


94 


GEOMETRY. 


[Th.   LII. 


Corollary  i.  All  diameters  go  through  the  centre,  and 
halve  each  other  there ;  conversely,  chords  halving  each 
other  are  diameters. 

Def.  Two  diameters  each  halving  all  the  chords  parallel 
to  the  other  are  called  conjugate. 

Corollary  2.  In  the  circle  two  diameters  normal  to  each 
other  are  conjugate  ;  and  conversely,  two  conjugate  diameters 
are  normal  to  each  other. 

N.B.  Other  curves,  as  Ellipse  and  Hyperbola,  have 
conjugate  diameters  not  in  general  normal  to  each  other 
(Fig.  73)- 


*ii3.    Theorem  LII. 

(Fig.  74). 


All  diameters  of  a  circle  are  equal 


Fig.  73.  Fig.  74. 

Proof.  Let  DR  and  D^R^  be  two  diameters.  The  figure 
DD'RR  is  a  parallelogram  (why?),  and  DV  is  parallel  to 
RR^ ;    hence  the  mid-normal  of  these  parallels  is  a  diameter 


Th.  LIL]  the    circle.  95 

through  the  centre  S ;    hence  SD  and  SD'  are  symmetric 
and  equal ;  hence  DR  =  D'R\     Q.  e.  d. 

JDef.  A  half-diameter,  from  centre  to  circle,  is  called  a 
radius. 

Corollary  i.  All  radii  of  a  circle  are  equal ;  or,  all  points 
of  a  circle  are  equidistant  from  the  centre. 

Corollary  2.  Every  parallelogram  inscribed  in  a  circle  is 
a  rectangle. 

N.B.  By  help  of  this  important  property  the  circle  is 
commonly  defined  as  a  plane  curve  all  points  of  which  are 
equidistant  fro7n  a  point  within  called  the  centre.  The  com- 
mon distance  of  all  points  of  the  circle  from  the  centre  is 
often  called  the  radius.  We  have  deduced  this  property 
from  the  homoeoidality ;  conversely ^  we  may  deduce  the 
homoeoidality  from  this  property  taken  as  definition.  But 
if  there  were  no  such  surface  as  the  plane,  at  least  for  our 
intuition,  the  circle  might  still  exist  on  the  sphere-surface, 
without  centre,  but  with  the  body  of  its  properties  unimpaired. 
Hence  it  seems  better  to  define  the  circle  by  its  intrinsic 
homoeoidality  than  by  its  extrinsic  centraHty. 

Corollary  i.  All  points  within  the  circle  are  less,  and  all 
points  without  are  more,  than  the  radius  distant  from  the 
centre. 

Defs.  The  two  symmetric  halves  into  which  a  diameter 
cuts  a  circle  are  called  semicircles.  The  part  of  the  plane 
bounded  by  an  arc  and  its  chord  is  called  a  segment ;  the 
part  bounded  by  an  arc  and  the  two  radii  to  its  ends  is 
called  a  sector.  If  the  sum  of  two  arcs  be  a  circle,  we  may 
call  them  explemental,  the  one  minor ^  the  other  major ; 
every  chord  belongs  equally  to  eacli  of  two  explemental  arcs, 
but  in  general,  unless  otherwise  stated,  it  is  the  minor  that 


96  GEOMETRY.  [Th.  LIII. 

is  referred  to.  Two  arcs  whose  sum  is  a  half-circle  are 
called  supplemental ;  two  whose  sum  is  a  quarter- circle  or 
quadrant  are  called  complemental. 

Corollary  2.  All  circles  of  the  same  radius  are  congruent ; 
also,  all  semicircles  of  the  same  radius  are  congruent,  and 
all  quadrants  of  the  same  radius  are  congruent. 

Corollary  3.  Any  circle  may  be  slipped  round  at  will 
upon  itself  about  its  centre  as  a  pivot,  Hke  a  wheel  about  its 
axle,  without  changing  in  the  least  the  position  of  the  whole 
circle. 

114.  From  the  foregoing  it  is  clear  that  if  we  hold  one 
point  of  a  ray  fixed,  and  turn  the  ray  in  the  plane  about  the 
fixed  point,  every  other  point  of  it  will  trace  out  a  circle 
about  the  fixed  point  as  a  centre.  An  instrument,  one  point 
of  which  may  be  fixed  while  the  other  is  movable  about  in  a 
plane,  is  called  a  compass  or  pair  of  compasses,  and  is  both 
the  simplest  and  the  most  important  of  all  instruments  for 
drawing. 

115.  Theorem  LIII.  —  Through  any  three  points  not  col- 
linear  one,  and  only  one,  circle  may  be  drawn. 

Proof.  Let  A,  B,  C  be  the  three  points  not  collinear 
(Fig.  75).  We  have  already  seen  that  the  mid-normals  to 
the  tracts  AB,  BC,  CA  concur  in  a  point  6*  equidistant  from 
A,  B,  and  C;  hence  a  circle  about  »S  with  radius  d  passes 
through  A,  B,  C.  Also  there  is  only  one  point  thus  equi- 
distant from  A,  B,  C  (why?)  ;  hence  there  is  only  one 
circle  through  A,  B,  C.     Q.  e.  d. 

Def.  The  circle  through  the  vertices  A,  B,  C,  of  a  A  is 
called  the  circum-circle  of  the  A. 

Corollary  i.  A  A,  or  a  triplet  of  points,  or  a  triplet  of 
rays,  determines  one,  and  only  one,  circle. 


Th.  LIIL] 


THE    CIRCLE. 


97 


Corollary  2.  Through  two  points,  A  and  B,  any  number 
of  circles  may  be  drawn.  Their  centres  all  lie  on  the  mid- 
normal  of  AB. 

Corollary  3.  As  BC  turns  clockwise  about  ^  as  a  pivot, 
the  intersection  S,  the  centre  of  the  circle  through  A,  B,  C, 
retires  upward  ever  faster  and  faster  along  the  mid-normal  N 
of  AB ;  when  C  becomes  collinear  with  A  and  B,  the  inter- 


FiG.  75. 

section  of  the  raid-normals  of  AB  and  BC  vanishes  from 
finity,  or  retires  to  infinity,  as  the  phrase  is.  As  BC  keeps 
on  turning,  S  reappears  in  finity  below  and  moves  slower  and 
slower  upward  along  the  mid-normal.  Moreover,  a  circle 
passes  through  A,  B,  and  C,  no  matter  how  close  C  may  lie 
to  the  ray  AB,  nor  on  which  side  of  it :  only  as  C  falls  upon 
the  ray  does  the  centre  of  the  circle  vanish  into  infinity ;  that 
is,  we  may  draw  a  circle  that  shall  fit  as  close  to  the  ray  AB 
as  we  please,  though  not  upon  it,  by  retiring  the  centre  far 


98  GEOMETRY.  [Th.   LIV. 

enough.  Hence  a  ray  may  be  conceived  as  a  circle  with 
centre  retired  to  infinity ;  it  is  strictly  the  limit  of  a  circle 
whose  centre  has  retired,  along  a  normal  to  it,  without  limit. 

ii6.  Theorem  LIV.  — A  circle  can  cut  a  ray  in  only  two 
points. 

For  there  are  only  two  points  on  a  ray  at  a  given  distance 
from  a  fixed  point  (why?),     q.  e.  d. 

117.  Theorem  LV.  —  Secants  that  make  equal  angles  with 
the  centre  ray  (or  axis)  through  their  intersection  intercept 
equal  arcs  on  the  circle. 

Proof.  For  both  the  two  semicircles  and  the  two  secants 
are  symmetric  as  to  the  axis  IS  (why?)  ;  hence,  on  folding 
over  the  one  half-plane  upon  the  other,  A  falls  on  A\  B  on 
B\  arc  a  fits  on  arc  a\  and  chord  c  on  chord  c^  (Fig.  76). 

Q.  E.  D. 


Th.  LVI.]  the   circle.  99 

Conversely,  Secants  that  intercept  equal  arcs  make  equal 
angles  with  the  axis  through  their  intersection. 

Proof.  Let  L  and  Z'  intersect  equal  arcs  AB  and  AB\ 
Draw  the  mid-normal  of  A  A ;  it  is  an  axis  of  symmetry 
(why?).  On  folding  over  the  left  half- plane  upon  the  right 
half-plane,  A  falls  on  A^  and  B  onB'  (why?)  ;  hence  AB 
and  A'B'  are  symmetric ;  hence  they  meet  on  the  axis  and 
make  equal  angles  with  it  (why?),     q.  e.  d. 

Corollary  i.  Equal  chords  are  equidistant  from  the  cen- 
tre ;  and  conversely ,  Chords  equidistant  from  the  centre  are 
equal. 

Corollary  2.  The  greater  of  two  unequal  chords  is  less 
distant  from  the  centre. 

Corollary  3.    A  diameter  is  the  greatest  chord. 

Corollary  4.  Arcs  intercepted  by  two  parallel  chords  are 
equal. 

Corollary  5.  Equal  chords  or  arcs  subtend  equal  central 
angles  (angles  at  the  centre),  and  conversely. 

Corollary  6.  Of  two  unequal  chords  or  arcs,  the  greater 
subtends  the  greater  central  angle. 

What  figure  is  determined  by  two  parallel  chords  and  the 
chords  of  the  intercepted  arcs  ?  By  two  secants  that  inter- 
cept equal  arcs  and  the  central  normals  thereto  ? 

118.  Theorem  LVI.  —  A  central  angle  sub te tided  by  a  cer- 
tain arc  (or  chord)  is  double  the  peripheral  angle  subtended 
by  the  same  (or  an  equal)  arc  (or  chord)  (Fig.  77). 

Proof.  Let  ASB  be  a  central  angle,  and  APB  be  a 
peripheral  angle  (periphery  =  circumference,  the  circle 
itself),  subtended  by  the  same  arc  or  chord  AB.     Draw  the 


100  GEOMETRY.  *  [Th.  LVI. 

diameter  PZ>.  Then  the  A  ASP  and  BSF  are  isosceles 
(why?)  ;  hence  the  angle  ASD  =  2  angle  AFD,  and  angle 
BSD  =  2  angle  BPD  (why  ?)  ;   hence  angle  ASB  =  2  angle 

APB.      Q.E.D. 

P ^ 


Fig.  77. 

Corollary  i .  All  peripheral  angles  subtended  by  (or  stand- 
ing on)  the  same  or  equal  chords  or  arcs  are  equal.  Hence, 
as  P  moves  round  from  A  to  B,  the  angle  APB  remains 
unchanged  in  size. 

Def.  An  angle  with  its  vertex  on  a  certain  arc,  and  its 
arms  passing  through  the  ends  of  that  arc,  is  said  to  be 
inscribed  in  that  arc.  Hence  for  an  angle  to  be  inscribed 
in  a  certain  arc,  and  for  it  to  stand  on  the  explemental  arc, 
are  equivalent. 

Corollary  2.  All  angles  inscribed  in  the  same  or  equal 
arcs  of  the  same  or  equal  circles  are  equal. 

Corollary  3.  As  the  vertex  Z'  of  a  peripheral  angle  sub- 
tended by  an  arc  (or  chord)  AB,  in  passing  round  a  circle 
goes  through  either  end  of  the  arc  (or  chord),  the  angle 
itself  leaps  in  value,  changes  to  its  supplement. 


Th.  LVIIL]  THE    CIRCLE.  101 

119.  Theorem  LVII.  —  The  locus  of  the  vertex  of  a  given 
angle  standing  on  a  given  tract  is  two  synu?ietric  circular 
arcs  through  the  ends  of  the  tract  (Fig.  78). 

\0 


Fig.  78. 

Proof.  Let  F  be  the  vertex  of  the  given  angle,  in  any 
position,  standing  on  the  tract  AB.  Through  A,  F,  and  B 
draw  a  circular  arc  subtended  by  AB.  We  have  just  seen 
that  as  long  as  F  stays  on  this  arc,  the  angle  F  remains  the 
same  in  size.  Moreover,  the  point  F  cannot  be  without  the 
arc,  as  at  O,  because  the  angle  A  OB  is  less  than  AFB 
(why?)  ;  neither  can  it  come  within  the  arc,  as  to  /,  because 
the  angle  AIB  is  greater  than  AFB  (why  ?)  ;  hence  so  long 
as  the  angle  is  constant  in  size  the  vertex  must  remain  on 
the  arc  AFB  or  on  its  symmetric  arc  AF^B^  of  which  plainly 
the  same  may  be  said.     q.  e.  d. 

120.  Theorem  LVIII.  —  The  angle  inscribed  in  a  semi- 
circle (or  standing  on  a  semicircle  or  diameter)  is  a  right- 
angle  (Fig.  79). 


102 


GEOMETRY. 


[Th.  LVIII. 


Proof.  Let  ^^C  be  any  angle  in  a  semicircle.  Then  it 
is  half  of  the  central  angle  ASC  (why?),  which  is  a  straight 
angle  (why?),     q. e. d. 


Fig.  79. 

Now  let  the  vertex  B,  the  intersection  of  the  rays  L  and 
N,  move  round  the  circle  toward  C ;  the  angle  ABC  re- 
mains a  right  angle,  no  matter  how  close  B  approaches  to  C ; 
moreover,  when  B  passes  C,  into  the  lower  semicircle,  the 
angle  remains  a  right  angle  (why?).  That  is,  the  angle  2itB 
remains  a  right  angle,  no  matter  from  which  side  nor  how 
close  B  approaches  to  C.  Hence  it  is  a  right  angle  even 
when  B  falls  on  C,  But  then  the  ray  L  falls  on  the  diame- 
ter A  C,  hence  the  ray  N  takes  the  position  T  normal  to  the 
diameter  (or  radius)  at  its  end.  Such  a  normal  to  a  radius 
at  its  end  is  called  a  tangent  to  the  circle  at  the  point  of 
tangence  (or  touch  or  contact^  C. 

Def.  A  ray  normal  to  a  tangent  to'a  curve  at  the  point 
of  touch  is  called  normal  to  the  curve  itself.     Hence 

Corollary.  All  radii  of  a  circle  are  normal  to  the  circle ; 
and  conversely,  all  normals  to  a  circle  are  radii  of  the  circle. 


Th.  LX.] 


THE    CIRCLE. 


103 


121.  Theorem  LIX.  —  All  points  on  a  tangent^  except  the 
point  of  contact,  lie  outside  of  the  circle. 

Proof.  For  the  point  of  touch  is  distant  radius  from  the 
centre  (why?),  and  all  other  points,  as  Z>,  of  the  tangent 
are  further  from  the  centre  (why?)  ;  hence  all  other  points 
of  the  tangent  are  without  the  circle  (why?),     q.  e.  d. 

122.  Theorem  LX.  —  The  point  of  tangence  is  a  double 
point. 

Proof.  For  it  is  on  a  diameter,  or  axis  of  symmetry,  of 
the  circle,  and  every  such  point  is  a  double  point  with 
respect  to  that  axis. 

Independently  of  this  consideration,  it  is  seen  that  the 
chord  CB  becomes  the  tangent  CT  when,  and  only  when, 
the  points  B  and  C  fall  together  in  C. 


Fig.  8o. 


Still  otherwise,  let  AB  be  any  chord  of  a  circle  about 
(Fig.  8o)  O.  Draw  the  mid-normal  OD.  Now  let  the 
circle  shrink  about  the  centre  O  :  the  points  A  and  B  move 


104 


GEOMETRY. 


[Th.  LXI. 


towards  each  other,  and  as  D  is  always  mid-way  between 
them  they  finally  fall  together  in  D,  and  their  join  is  tan- 
gent at  D  to  the  circle  of  radius  OD. 

Def.  Two  points  thus  falUng  together  in  a  double  point 
are  called  consecutive  points.  Accordingly  we  may  define 
a  tangent  to  a  circle  (or  to  any  curve)  as  a  ray  through  two 
consecutive  points  of  the  circle  (or  curve).  Adopting  this 
definition,  let  the  student  prove 

123.  Theorem  LXI.  Every  tangent  to  a  circle  is  normal 
to  a  radius  at  its  end ;  conversely,  Every  normal  to  a  radius 
at  its  end  is  tangent  to  the  circle. 

124.  Theorem  LXII.  The  angle  between  a  tangent  and 
a  chord  equals  the  peripheral  angle  on  the  same  chords  or 
equals  half  the  angle  of  the  chord  (Fig.  81). 


Proof.  For  if  DT  be  a  diameter,  then  the  angles  BDT 
and  BTA  are  equal,  being  complements  of  the  same  angle 
BTD  (why?).     Or  thus  :   TB 've>  d.  chord,  and  TA  is  also  a 


Th.   LXIV.] 


THE    CIRCLE. 


105 


chord,  through  the  double  point  T ;  hence  the  angle  BTA 
is  a  peripheral  angle  standing  on  the  arc  TB.     q.  e.  d. 

125.  Theorem  LXIII.  —  The  angle  between  two  secants 
is  half  the  sum  or  half  the  difference  of  the  angles  of  the 
intercepted  arcs,  according  as  the  secants  intersect  within  or 
without  the  circle. 

Proof.  For  on  drawing  AB^  the  angle  /  is  seen  (Fig. 
82)  to  be  the  sum,  and  the  angle  O  the  difference,  of  the 


Fig.  82. 
angles  at  A  and  B^  standing  on  the  arcs  AA'  and  BB\ 

Q.  E.  D. 

126.  Theorem  LXIV. — An  encyclic  quadrangle  has  its 
opposite  angles  supplemental. 

Proof.  For  the  angles  B  and  D  are  halves  of  the  two 
central  angles  ASC  and  CSA,  whose  sum  is  a  round  angle. 
Hence  the  sum  of  B  and  Z>  is  a  straight  angle,     q.  e.  d. 


106  GEOMETRY.  [Th.  LXV. 

127.    Theorem  LXV.  —  Conversely,  A    quadrangle  with 
its  opposite  angles  supplemental  is  encyclic  (Fig.  ^2)). 


Fig.  83. 

Proof.  Let  ABCD  be  the  quadrangle  with  the  angles 
A  and  C,  ^  and  D,  supplemental.  About  the  A  ABC  draw 
a  circle.  If  P  be  any  point  on  the  arc  of  this  circle  exple- 
mental  to  ABC,  then  the  angle  APC  xs^  the  supplement  of 
ABC  \  but  if  P  be  not  on  this  arc,  then  the  angle  A  PC  is 
either  greater  or  less  than  that  supplement  (why?).  Now 
the  angle  D  is  that  supplement ;  hence  D  is  on  the  arc. 

Q.  E.  D. 

128.    Relations  of  circles  to  each  other. 

Suppose  two  circles  K  and  K^  of  radii  r  and  r'  to  be 
concentric,  i.e.  to  have  the  same  centre  O.  Then,  plainly, 
the  distance  between  them  measured  on  any  half-axis  OR 
is  r—r\  the  difference  of  the  radii.  Draw  tangents  AT, 
A^T',  where  00^  cuts  the  circles.  They  are  parallel  (why?). 
Now  let  the  centre  of  K'  move  out  on  6>(9'  a  distance  r  —  /; 
then  ^  falls  on  A^  and  A'T^  on.  AT;  the  circles  have  a 
common  tangent  at  A  and  are  said  to  touch  each  other 
innerly  at  ^  (Fig.  84). 


Th.  LXVIL] 


THE   CIRCLE. 


107 


Now  let  (9'  move  still  further  along  00^ ;  then  the  circles 
will  lie  partly  within,  partly  without,  each  other ;  they  will 
intersect  at  two  points,  and  only  two  (why?),  symmetric  as 
to  00^  (why?),  namely  /'and  P ;  hence 


"Tk'  i<' 


yfe) 


Fig.  84. 


Theorem  LXVI.  —  The  cotnmon  axis  of  two  circles  is  the 
mid-7iormal  of  their  conwion  chord. 

When  O^  is  distant  r  +  /  from  O,  the  circles  He  without 
each  other,  but  still  have  a  common  tangent  (why  ?)  and  are 
said  to  touch  outerly. 

As  O^  moves  still  further  away  from  O,  the  circles  cease  to 
touch  and  henceforth  He  entirely  without  each  other. 

Thus  we  find  that  there  are  three  critical  positions  depend- 
ing on  the  distance  d  between  the  centres  O  and  O^ : 

d=o,  when  the  circles  are  concentric. 

d=z  r—  r\  when  the  circles  touch  innerly. 

d=  r-\-  r',  when  the  circles  touch  outerly. 
There  are  also  three  intermediate  positions  : 
For  o  <  d  <  r—r^  the  one  circle  is  withi?i  the  other. 
For  r—r^<.  d<i  r  -\-  r  the  circles  intersect. 
For  r  -\-  r  <.  d  <.  ^  the  circles  lie  7vithout  each  other. 

129.  Theorem  LXVII.  —  From  any  point  without  a  circle 
two,  and  only  two,  tangents  may  be  drawn  to  the  circle  (Fig. 

85). 


108 


GEOMETRY. 


[Th.  LXVII. 


Proof.  Let  O  be  the  centre  of  the  circle  K^  and  P  be 
the  point  without.  On  OP  as  a  diameter  draw  a  circle  K^ ; 
only  one  such  circle  is  possible  (why?),  and  it  cuts  X  in  two, 
and  only  two,  points,  rand  T'.  Draw /^r and  PT' :  they 
are  tangent  to  ^  at  T  and  T'  (why  ?) .  Moreover,  no  other 
ray  through  P,  as  PC/,  is  tangent  to  X,  because  OUP  is  not 
a  right  angle  (why  ?) .     q.  e.  d. 


Fig.  85. 

De/.  The  chord  TT'  through  the  points  of  contact  of  the 
tangents  is  called  the  chord  of  contact  for  the  point  P  or 
the  polar  of  the  pole  P  (see  Art.      ) . 

The  angle  between  the  tangents  to  two  curves  at  the 
intersection  of  the  curves  is  called  the  angle  between  the 
curves  themselves.  When  this  is  a  right  angle,  the  curves 
are  said  to  intersect  orthogonally. 

The  distance  PT  or  PT'  is  called  the  tangent-length 
from  P  to  the  circle. 

Corollary  i .  Two  circles,  one  having  as  radius  the  tangent- 
length  from  its  centre  to  the  other,  intersect  orthogonally. 

Corollary  2.  Two  tangents  are  symmetric  as  to  the  axis 
through  their  intersection ;  hence,  also,  the  tangent-lengths 
are  equal. 


Th.  lxix.]  the  circle.  109 

130.  Theorem  LXVIII. — All  tangent-lengths  to  a  circle 
frofn  points  on  a  concentric  circle  are  equals  and  intercept 
equal  arcs  of  the  circle  (Fig.  %(i). 


Fig.  86. 

Proof.  For  \i  P  be  any  point  without  the  circle  K\  we 
may  turn  P  round  about  the  centre  (9  on  a  concentric  circle 
K^  without  affecting  any  of  the  relations  obtaining  (why  ?) . 

Or  thus  :  the  right  A  TOP  and  T  OP^  are  plainly  con- 
gruent (why?);  hence  PT=^  P^T  (why?),     q.e.d. 

*i3i.  Theorem  LXIX.  —  The  intercept  between  two  fixed 
tangents  on  a  third  tangent  subtends  a  constant  central  angle 
(Fig.  87). 

Proof.  Let  PT  and  PV  be  the  fixed  tangents,  FF'  the 
intercept  on  the  variable  ray  tangent  at  U.  Then  TPV  is 
a  constant  angle,  and  VOV^  is  half  of  TOT  (why?),  and 
hence  is  constant,     q.  e.  d. 


no 


GEOMETRY, 


[Th.  LXX. 


\  Fig.  87. 

132 .  Theorem  LXX.  — If  the  central  (or  peripheral)  angles 
of  the  common  chord  of  two  intersecting  circles  be  equal,  the 
circles  are  equal. 

Let  the  student  conduct  the  proof  suggested  by  the  figure 
(Fig.  ^^)j  and  let  him  prove  the  converse. 


Fig 


*i33.  Theorem  LXXI.  —  The  circum circle  of  a  A  equals 
the  circumcircle  of  the  orthocentre  and  any  two  vertices  of 
the  A  (Fig.  89). 

Proof.  Let  K  be  the  circumcircle  of  the  A  ABC,  K^  the 
circumcircle  of  A,  B,  and  O  the  orthocentre.     The  angles 


Th.  LXXIL] 


THE    CIRCLE. 


Ill 


Cand  B^ OA'  are  supplemental  (why?)  ;  also  the  angles  D 
and  BOA  are  supplemental  (why?)  ;  and  the  angles  BOA 
and  B'OA'  are  equal  (why?)  ;  hence  the  angles  Z>  and  C 
are  equal ;  hence  A'  =  A''  ( why  ?)     q.  e.  d. 


Fig.  89. 

*i34.  Theorem  LXXII. — T/ie  mid-points  of  the  sides  of  a 
A,  the  feet  of  its  altitudes,  and  the  7tiid-points  between  its 
07-thocentre  and  vertices^  are  nine  encyclic  points. 

Proof.  Let  a  circle  through  X,  V,  Z,  the  mid-points  of  the 
sides,  cut  the  sides  in  three  other  points,  O]  V,  W.  Then 
the  angle  ZXY=  angle  A  (why?),  and  also  =  angle  ZVY 
(why  ?)  ;  therefore  the  A  AZV  is  symmetrical.  Hence  the 
A  ZVB  is  also  symmetrical,  Z  is  equidistant  from  A,  V,  and 
By  and  the  angle  AVB  is  a  right  angle  (why  ?)  ;  so  also  the 
angles  at  C^and  IV;  i.e.  the  circle  through  the  mid-points  of 
the  sides  goes  through  the  feet  of  the  altitudes  (Fig.  90). 

Again,  if  the  circle  cuts  the  altitudes  at  P,  Q,  R,  then  the 
angle  F/'^=  angle  VZIV (why?)  =  2  angle  F^^(why?). 
Moreover,  A,  F,  O,  IV,  are  encyclic  (why?)  ;  hence  AO  is 
a  diameter  of  the  circle  through  them  (why?)  ;  and  VAW 
is  a  peripheral  angle  standing  on  the  arc  VIV;  hence  the 


112 


GEOMETRY. 


[Th.   LXXIII. 


double  angle  VjRJVmust  be  the  central  angle  of  the  same 
arc ;  t.e.  P  is  the  mi^-point  between  a  vertex  and  orthocen- 
tre  :  so,  also,  are  Q  aftd  R,  similarly,     q.  e.  d. 


Def.  This  remarkable  circle  is  called  the  9-point  circle, 
or  circle  of  Feuerbach,  of  the  A  ABC, 

Corollary.  The  radius  of  the  9-point  circle  is  half  the 
radius  of  the  circumcircle. 

135-  Def.  A  Polygon  all  of  whose  sides  touch  a  circle  is 
said  to  be  circumscribed  about  it,  and  the  circle  is  said  to  be 
inscribed  in  the  polygon. 

Theorem  LXXIII.  — A  circle  may  be  inscribed  in  any  A. 

Proof.  Let  ABC  be  any  A  (see  Fig.  59).  Draw  the 
inner  mid-rays  of  the  angles  at  A,  B,  C ;  they  concur  in  the 
in-centre  /of  the  A,  equidistant  from  the  three  sides  (why?). 
About  this  point  as  centre  with  this  common  distance  as 
radius  draw  a  circle ;  it  will  touch  the  three  sides  of  the  A 
(why  and  where  ?) .     q.  e.  d. 

N.B.  We  have  seen  that  the  outer  mid-rays  of  the  angles 
concur  in  pairs  with  the  inner  mid-rays  of  the  angles  in  the 
three  ex-centres  £1,  £2,  -^sl  also  equidistant  from  the  sides 


Th.  LXXIV.] 


THE    CIRCLE. 


113 


(Fig.  60).  The  circles  about  these  touch  only  two  sides 
innerly,  but  the  third  side  outerly,  ai^  hence  are  called 
escribed,  or  ex-circles.  • 

Corollary,    Four,  and  only  four,  circles  touch,  each,  all  the 
sides  of  a  A. 


135  a.  Theorem  LXXIV.  — In^  a  4-side  circumscribed 
about  a  circle  the  sums  of  the  two  pairs  of  opposite  sides  are 
equal  (Fig.  91). 


Fig.  91. 

Proof.     The  sum  of  the  four  sides  is  plainly  2/4-2/^+22^ 
-f-  2Z£/,  and  the  sum  of  either  pair  of  opposites  is  t-\-u-\-v-\-w. 

Q.  E.  D. 

Conversely,  If  the  sums  of  two  pairs  of  opposite  sides  of  a 
4-side  be  equals  the  4-side  is  circumscribed  about  a  circle. 

Proof.     Let  two  counter  sides,  AB  and  DC  meet  in  /, 
and  inscribe  a  circle  K  in  the  triangle  ADL     Through  B 


114 


GEOMETRY. 


[Th.  LXXV 


draw  a  tangent  (Fig.  92)  to  A' at  U,  and  let  it  cut  DI  dX  C 
Then  since  ABCD  is  circumscribed  about  K,  we  have 


or 


Fig.  92. 

AB-\-CD  =  BC-\-DA. 
Also  AB+CD  =BC  +  DA  (why ?). 

Whence  CD-CD^BC-B  C\ 

CC  =  BC-BC. 


Hence  Cand  C  fall  together  (why?     Art.  56).     q. e.  d. 

136.  Theorem  LXXV.  —  The  tangent-length  fro?n  a  ver- 
tex of  a  /\  to  the  in-circle  equals  half  the  perimeter  of  the  A 
less  the  opposite  side  (Fig.  93). 


Th.   LXXVI.] 


THE    CIRCLE. 


115 


Proof.  For  the  sum  of  CE  ^  CD -\-  BD  +  BF  is  plainly 
2a  (why?)  ;  subtract  this  from  the  whole  perimeter,  a  -{-  b  -\-  c^ 
and  there  remains  AE  -\-  AF—  a  -\-b  -\-  c  —  2a^  or  AE  = 
b  -\-  c—  a        .  E, 

—^ =  AF.       Q.  E.  D. 


FIG.  93.  ; 

It  is  common  and  convenient  to  denote  the  perimeter 
(Fig.  93)  (=  measure  round  =  sum  of  sides)  by  2J-;  then 
we  see  that  the  tangent-lengths  from  A^  B,  C,  are  s  —  a^ 
s  —  bj  s  —  c. 

Corollary.  The  tangent-length  from  any  vertex,  A,  of  a 
A  to  the  opposite  ex-circle  and  the  two  adjacent  ex-circles 
are  s,  s—b,  s  —  c.  Hence  s  —  a,  s,  s  —  b,  s  —  c,  are  the 
four  tangent-lengths  from  any  vertex,  ^,  of  a  A  to  the  in- 
circle  and  the  three  ex-circles. 

These  relations  are  useful  and  important. 

137.    Theorem  LXXVI.  —  There  is  a  regular  n-side. 
Proof.     For  the  angle  is  a  continuous  magnitude  (why?)  ; 
hence  there  are  angles  of  all  sizes  from  zero  to  a  round 


116 


GEOMETRY. 


[Th.  LXXVI. 


angle  ;   hence  there  is  an  angle,  the  -  part  of  a  round  angle, 

n 

such  that,  taken  n  times  in  addition,  the  sum  will  be  a  round 
angle.  Suppose  such  an  angle  drawn,  whether  or  not  we 
can  actually  draw  it,  and  suppose  n  such  angles  placed  con- 
secutively around  any  point  O,  so  as  to  make  a  round  angle. 
In  other  words,  suppose  n  half-rays  drawn  cutting  the 
round  angle  about  O  into  n  equal  angles.  Draw  a  circle 
about  O,  with  (Fig.  94)  any  radius,  and  draw  the  7?  chords 


Fig.  94. 

subtending  the  n  equal  central  angles.  These  chords  are 
all  equal  (why?),  and  subtend  equal  arcs,  and  they  form  an 
;/-side.     Moreover,  the  angle  between  two  consecutive  sides 


Th.  lxxviil]  the  circle.  117 

is  constant  in  size,  because  it  stands  on  the  ^^  ~  ^  part  of 

tlie  circle.     Hence  the  «-side  is  both  equilateral  and  equian- 
gular ;  that  is,  it  is  regular,     q.  e.  d. 

Corollary.     The   inner  angle  of  a  regular  «-side  is  the 

part  of  a  straight  angle.  \ 


(^") 


Find  the  value  in  degrees  of  the  inner  angles  of  the  first 
ten  regular  «-sides. 

N.B.  The  foregoing  demonstration  merely  ; settles  the 
question  of  the  existence  or  logical  possibility  of  the  regular 
;/-side.  The  problem  of  actually  drawing  such  a  figure  is 
one  of  the  most  intricate  in  all  mathematics,  and  has  been 
solved  only  for  certain  very  special  classes  of  values  of  n. 
But  in  order  to  discover  the  properties  of  the  figure,  it  is  by 
no  means  necessary  to  be  able  to  draw  it  accurately.  It  is 
only  since  1864  that  we  have  known  how  to  draw  a  straight 
line  or  ray  exactly. 

137  a.'  Theorem  LXXVII.  —  The  vertices  of  a  regular 
n-side  are  encyclic  (Fig.  94). 

ftroof.  Through  any  three  vertices,  as  A,  B,  C,  of  a  regular 
;2-side,  draw  a  circle  K ;  about  C  with  radius  CB  draw 
another  circle.  The  fourth  vertex  D  must  lie  on  this  circle 
(why?).  If  it  lie  on  the  circle  K,  then  the  angle  BCD  — 
angle  ABC,  as  is  the  case  in  the  regular  «-side.  Neither 
can  it  lie  off  of  K,  as  at  Z>'  or  D",  because  then  the  angle 
BCD'  or  BCD''  would  not  equal  angle  BCD  (why?),  and 
hence  would  not  equal  angle  ABC.  Hence  the  next  vertex 
must  lie  on  the  same  circle  K,  and  so  on  all  around,    q.  e.  d. 

138.  Theorem  LXXVIII.  —  The  sides  of  a  regular  n-side 
are pericyclic  (that  is,  they  all  touch  a  circle). 


118 


GEOMETRY. 


[Th.  LXXIX. 


Proof.    For,  on  drawing  the  radii  of  the  circumcircle  K 
(Fig.  95)  to  the  vertices,  we  get  n  congruent  symmetric  A 
A 


O  Fig.  95. 

(why?).  The  altitudes  of  all  are  the  same  (why?)  ;  with 
this  common  altitude  as  radius  draw  another  circle,  K\ 
about  the  same  centre.  It  will  touch  each  of  the  sides 
(why?).     Q.  E.  D. 

Corollary.  The  points  of  touch  of  the  sides  of  the  regular 
circumscribed  /z-side  are  mid-points  of  the  sides. 

139.  Theorem  LXXIX.  —  The  points  of  touch  of  a  regular 
circumscribed  n-side  are  the  vertices  of  a  regular  inscribed 
n-side. 

Proof.  Connect  the  points  of  touch  consecutively.  Then 
the  A  so  formed  are  all  congruent  (why?)  ;  hence  the 
joining  chords  are  equal;  hence  the  arcs  are  equal; 
hence  the  Theorem,     q.  e.  d. 


CIRCLE  AS  ENVELOPE.  119 

THE   CIRCLE  AS   ENVELOPE. 

*i4oa.  Thus  far  we  have  regarded  the  circle  from  various 
points  of  view ;  from  the  most  familiar  it  was  seen  to  be  the 
locus  of  a  point  in  a  plane  at  a  fixed  distance  from  a  fixed 
point.  An  almost  equally  important  conception  of  the  curve 
treats  it  not  as  the  locus  of  a  point,  but  as  the  envelope  of  a 
ray.  If  the  point  P  moves  in  the  plane  always  equidistant 
from  (9,  then  its  locus  is  the  circle,  on  which  it  may  always 
be  found  ;  also,  if  the  ray  R  moves  about  in  the  plane  always 
equidistant  from  6>,  then  its  envelope  is  the  circle,  on  which 
it  may  always  be  found,  on  which  it  Hes,  which  it  continually 
touches.  The  point  traces  the  circle,  the  ray  envelops  the 
circle,  which  is  accordingly  called  the  envelope  (i.e.  the 
enveloped  curve  —  French  enveloppee)  of  the  ray.  In  higher 
mathematics  the  notion  of  the  ray,  instead  of  the  point,  as 
the  determining  element  in  the  nature  of  a  curve,  attains 
more  and  more  significance.  In  this  text  we  are  confined 
to  the  circle  —  the  envelope  of  a  ray  in  a  plane,  at  a  fixed 
distance  from  a  fixed  point. 

*i4ob.  It  is  not  only  rays,  however,  that  may  envelop  a 
curve ;  but  circles,  and  in  fact  any  other  curves.  Thus,  let 
the  student  draw  a  system  of  equal  circles,  having  their 
centres  on  another  circle ;  the  envelope  will  at  once  be  seen 
to  be  a  pair  of  concentric  circles.  Let  him  also  find  the 
envelope  of  a  system  of  circles  equal  and  with  centres  on  a 
given  ray.  In  general,  let  him  find  the  envelope  of  a  circle 
whose  centre  moves  on  any  given  curve.  Lastly,  let  him 
draw  a  large  number  of  circles  all  of  which  pass  through  a 
fixed  point,  while  their  centres  all  lie  on  a  fixed  circle,  and 
let  him  observe  what  curve  they  shadow  forth  as  envelope. 

Show  that  as  the  pole  of  a  chord  (or  ray)  traces  a  circle. 


120  GEOMETRY. 

the  chord  itself  envelops  a  concentric  circle,  and  con- 
versely. 

Show  that  tangents  from  two  points  on  a  centre  ray  form 
a  kite,  and  conversely.  Also  the  chords  of  contact  are 
parallel,  and  conversely. 

O  is  the  centre  of  a  circle,  P  any  point  without  it.  Show 
how  to  find  the  point  of  touch  of  the  tangents  from  P^  by 
drawing  a  circle  about  O  through  P  and  a  tangent  where 
OP  cuts  the  given  circle. 

CONSTRUCTIONS. 

140.  Hitherto,  in  our  reasoning  about  concepts,  figures 
have  not  been  at  all  necessary,  though  exceedingly  useful  in 
making  sharp  and  precise  our  imagination  of  the  relations 
under  consideration,  in  furnishing  sensible  examples  of  the 
highly  general  notions  that  we  dealt  with.  The  conclusions 
reached  thus  far  all  He  wrapt  up  in  axioms  and  in  our  defi- 
nitions of  point,  ray,  and  circle,  and  our  work  has  been  one 
of  explication  only ;  we  have  merely  brought  them  forth  to 
light.  Our  demonstrations  have  not  presumed  ability  to  draw 
accurately,  and  would  remain  unshaken  if  we  could  not  draw 
at  all.  Nevertheless,  for  many  practical  purposes,  it  is  ex- 
tremely important  and  even  indispensable  that  we  actually 
make  the  constructions  and  draw  the  figures  that  thus  far  we 
have  merely  supposed  made  and  drawn. 

141.  What  is  meant  by  drawing  a  ray,  circle,  or  any  line? 
Any  mark,  whether  of  ink  or  chalk,  though  a  solid,  may  be 
treated  as  a  line  by  abstraction.  Only  its  length,  not  its 
width  nor  thickness,  concerns  us.  How  to  make  not  just 
any  mark,  but  some  particular  mark  called  for,  is  our  prob- 
lem  {Trpo/SXrjiJLa  =  anything  thrown  forward  as  a  task),  and 


CONSTRUCTIONS.  121 

its  solution  consists  accordingly  of  two  parts,  the  logical  and 
the  mechanical.  The  first  is  accomphshed  by  fixing  exactly 
in  thought  the  position  of  all  the  geometric  elements  (points, 
rays,  circles)  in  question  ;  the  second,  by  making  marks  that 
by  abstraction  may  be  treated  as  these  elements.  Now,  a 
point  is  fixed  as  the  join  of  two  rays,  a  ray  as  the  join  of  two 
points  (by  what  axiom?)  ;  a  circle  is  fixed  or  determined  by 
its  centre  and  radius  (why?),  or  by  three  points  on  it  (why?). 
Accordingly,  when  we  know  two  rays  through  a  point,  or  two 
points  on  a  ray,  or  centre  and  radius,  or  three  points  of  a 
circle,  we  know  the  point,  or  ray,  or  circle  completely.  The 
logical  part  of  our  work  is  finished,  then,  when  we  determine 
every  point  as  the  join  of  two  known  rays,  every  ray  as  the 
join  of  two  known  points,  every  circle  as  drawn  through  three 
known  points  or  about  a  known  centre  with  a  known  radius. 
The  mechanical  part  of  the  solution  requires  us  to  put  and 
keep  a  point  in  motion  along  a  circle  or  a  ray.  Circular 
motion  is  brought  about  by  the  compasses  already  described 
(Art.  114),  of  which  the  shape  is  arbitrary,  the  necessary 
parts  being  merely  a  fixed  point  rigidly  connected  in  any  way 
with  a  movable  point.  But  in  the  ruler  one  edge  is  supposed 
made  straight  to  begin  with,  so  that  a  pencil  point  gliding 
along  it  may  trace  a  straight  mark.  Hence  the  use  of  the 
ruler  is  really  illogical,  since  it  assumes  the  problem  of  draw- 
ing a  ray  or  straight  line  as  already  solved  in  constructing 
the  straight  edge.  To  say  that,  in  order  to  draw  a  straight 
line  J  we  must  take  a  straight  edge  and  pass  a  pencil  point 
along  it,  is  no  better  logically  than  to  say  that,  in  order  to 
draw  a  circular  line,  we  must  take  a  circular  edge  and  pass 
a  pencil  point  along  it.  The  question  at  once  arises.  How 
make  the  edge  straight  or  circular  in  the  first  place  ?  It  was 
not  until  1864  that  PeaucelHer  won,  though  he  did  not  at 
once  receive,  the  Montyon  prize  from  the  French  Academy 


122  GEOMETRY. 

by  solving  the  thousand-year-old  problem  of  imparting  rec- 
tilinear motion  to  a  point  without  guiding  edge  of  any  kind 
(Page  ooo).  But,  though  the  ruler  is  logically  valueless,  it  is 
practically  invaluable,  even  after  the  great  discovery  of  Peau- 
cellier.  Its  edge  being  assumed  as  straight  and  of  any 
desired  length,  and  a  pair  of  compasses  of  adjustable  size 
being  given,  we  now  make  the  following  Postulates : 

I.  About  any  point  may  be  drawn  a  circle  of  any  radius, 

II.  Through  any  two  points  may  be  drawn  a  ray  (more 
strictly,  a  tract  of  any  required  length) . 

Corollary.  On  any  ray  from  any  point  on  it  we  may  lay 
off  a  tract  of  any  required  length. 

These  are  the  only  instruments  used  or  postulates  assumed 
in  the  constructions  of  Elementary  Geometry. 

142.  The  fundamental  relations  of  rays  to  each  other  are 
two  :  Normality  and  Parallelism.     Hence 

Problem  I.  —  To  draw  a  ray  normal  to  a  give  ft  ray.  Since 
there  are  many  rays  normal  to  a  given  ray,  to  make  the 
problem  definite  we  insert  the  Hmiting  condition,  through  a 
given  point.     Two  cases  then  arise  : 

A.  When  the  given  point  is  on  the  given  ray.  All  we  can 
do  is  to  draw  a  circle  about  the  point  P.  It  cuts  the  ray  at 
two  points,  A  and  A\  symmetric  as  to  F.  Hence  the  mid- 
normal  of  AA^  is  the  normal  sought.  Hence  any  point  on 
this  normal  Hes  on  two  circles  of  equal  radius  about  A  and 
A\     Hence  (Fig.  96) 

Solution.  From  the  given  point  P  lay  off  on  the  given 
ray  two  equal  tracts  PA,  PA'.  About  A  and  A'  draw  two 
equal  circles.  Through  their  points  of  intersection  draw 
their  common  chord.     It  is  the  normal  sought. 


CONSTR  UCTIONS. 


123 


Proof.  For  it  is  the  mid-normal  of  AA\  since  it  has  two 
of  its  points  equidistant  from  A  and  A\  and  P  is  the  mid- 
point of  AA\ 


Fig.  96. 


Query :  What  radius  shall  we  take  for  the  circles  about 
^and^'? 

B.  When  the  given  point  is  not  on  the  given  ray.  All  we 
can  do  is  to  draw  a  circle  about  the  given  point  P.  Let  it 
cut  the  ray  at  A  and  A\  Then  the  mid-normal  to  A  A'  is 
the  normal  required  (why?).     Hence  (Fig.  97) 

Solution.  Determine  the  points  A,  A^  on  the  ray  by 
a  circle  about  the  given  point  P;  then  proceed  as  in  the 
first  case  (A). 


124 


GEOMETRY. 


Proof.     For    the    mid-normal    of  A  A   goes   through   P 
(why?). 


/K 


Fig;  97. 
Query :    What  radius  shall  \^^e  take  for  the  circle  about  P  ? 

143.  Problem  II.  —  To  dmw  a  parallel  to  a  given  ray. 
Since  there  are  many  parallels  to  every  ray,  to  make  the 
problem  definite  we  must  insert  the  limiting  condition, 
through  a  given  point ;  then  it  becomes  perfectly  definite 
(why?).  Manifestly  the  point  must  be  not  on  the  ray 
(why?).  We  now  reflect  that  a  transversal  makes  equal 
corresponding  angles  with  parallels,  and  we  have  just  learned 
to  draw  a  normal  transversal.     Hence  (Fig.  98) 

Solution.  Through  the  point  draw  a  normal  to  the  ray; 
through  the  same  point  draw  a  normal  to  this  normal.  It 
will  be  the  parallel  required. 

Proof.  For  it  goes  through  the  point  and  is  parallel  to  the 
ray  (why?). 

These  two  problems  have  been  discussed  at  such  length 
as  being  the  hinges  on  which  nearly  all  others  turn.      At 


CONSTRUCTIONS.  125 

the  end  of  a  problem  is  sometimes  written  Q.  E.  f.  =  quod 
erat  faciendum  =1  which  was  to  be  done,  and  translates  the 
Euclidean  OTrcp  cSet  Trpa^ai. 


\/l 


TX  ^ TX 


Fig.  98. 

144.  Problem  III.  —  To  bisect  a  given  tract,  or  to  draw 
the  mid-normaJ  to  a  given  tract,  AB, 

Proceed  as  in  Problem  I. 

145.  Problem  IV.  —  To  bisect  a  given  angle. 

Solution.  About  the  vertex  draw  any  circle  cutting  the 
arms  at  A  and  A\  and  draw  the  mid-normal  of  AA\  It  is 
the  mid-ray  sought  (why  ?) . 

Corolla?'}'.    Show  how  to  bisect  any  circular  arc  AB. 

146.  Problem  V.  —  To  bisect  the  angle  between  tivo  rays 
whose  join  is  not  given  (Fig.  99). 


126 


GEOMETRY. 


We  reflect  that  the  join  AA^  of  two  corresponding  points 
on  the  rays  makes  equal  angles  with  the  two  rays  that  form 
the  angle.     Hence 

Solution.  From  any  point  P  oi  L  draw  the  normal  to  it, 
cutting  M  at  Q.     From  Q  draw  the  normal  to  M.     Bisect 


Fig.  99. 

the  angle  at  Q  between  these  two  normals  by  the  mid-tract 
QR.  Draw  the  mid-normal  of  QR.  It  is  the  mid-ray 
sought  (why?). 

147.    Problem  VI.  —  To  multisect  a  given  tract  AB  (Fig. 
100). 

L 


B 

Fig.  icxj. 


CONSTR  UCTIONS. 


127 


Solution.  Through  either  end  of  the  tract,  as  A^  draw  any 
ray,  and  lay  off  on  it  from  A  successively  n  equal  tracts,  Z 
being  the  end  of  the  last.  Draw  BL.  Through  the  ends 
of  the  equal  tracts  draw  parallels  to  BL.  They  cut  AB 
into  n  equal  parts  (why  ?) . 

148.  Problem  VII.  —  To  draw  an  angle  of  given  size,  i.e. 
equal  to  a  given  angle  (Fig.  loi). 


,  /' 


y  I 


Solution.  At  any  point  A  of  either  arm  of  the  given  angle 
O  erect  a  normal  to  OA  cutting  the  other  arm  at  B.  From 
any  point  O  on  any  other  ray  lay  off  O^A'  =  OA,  and  normal 
to  the  ray  erect  A'B'  =  AB  and  draw  O'B'.  Then  angle  O' 
=  angle  O  (why  ?) . 

When  does  this  construction  fail  ?     How  proceed  then  ? 

149.  Problem  VIII.  —  To  draw  a  tract  of  given  length 
subtending  a  given  angle  and  parallel  to  a  given  ray. 

Data :  O  the  given  angle,  L  the  ray,  a  the  length  (Fig. 
102). 

Solution.  Through  any  point  P  of  either  arm  of  the  angle 
draw  a  parallel  to  the  ray,  and  lay  off  on  it  towards  the  other 


128 


GEOMETRY. 


arm  a  tract  PA  of  the  given  length  a.  Through  A  draw  a 
parallel  to  OP,  cutting  the  other  angle  arm  at  Q ;  through  Q 
draw  a  parallel  to  PA  meeting  OP2X  R.  QR  is  the  subtense 
sought  (why?). 


Fig.  102. 

150.    Problem  IX.  — To  construct  a  A  : 
A.    When  alternate  parts  (three  sides  or  three  angles)  are 
given. 

Solution.  About  the  ends  of  one  side  AB,  with  the  other 
sides  for  radii,  draw  circles  meeting  in  C.  Then  ABC  is 
the  A  sought  (why?)  (Fig.  103). 


Fig.  103. 

How  many  such  A  may  be  drawn  on  the  same  base  AB'> 
How  are  they  related?    When  is  the  solution  impossible? 


CONS  TR  UCTIONS. 


129 


When  the  angles  are  given,  apply  the  construction  of  Problem 
VII.     How  many  solutions  are  possible  ?     What  kind  of  A  ? 

B.  When  three  consecutive  parts  (two  sides  and  included 
angle  or  two  angles  and  included  side)  are  given. 

Solution.    Apply  the  construction  in  Problem  VII. 

C.  When  opposite  parts  (two  angles  and  an  opposite  side 
or  two  sides  and  an  opposite  angle)  are  given. 

Solution.  Apply  the  construction  in  Problem  VII.  When 
is  the  construction  ambiguous? 

D.  When  two  sides  and  the  altitude  to  the  third  side  are 
given. 

Solution.  Through  one  end  of  the  altitude  draw  a  normal 
to  it  for  the  base  ;  about  the  other  end  C  as  centre,  with  the 
sides  as  radii,  draw  circles  cutting  the  base  at  A  and  A\ 
B  and  B' ;  then  ACB  or  A'CB'  is  the  A  required.     Why? 

E.  When  two  sides  and  the  medial  of  the  third  side  are 
given. 

A 


m 


Fig.  104. 


130  GEOMETRY, 

\i  SA  be  the  medial  of  BC,  and  SA^  be  symmetric  with 
(Fig.  104)  SA  as  to  S,  then  ABA'C  is  a  parallelogram 
(why?)  ;  hence 

Solution.  Take  a  tract  the  double  of  the  medial.  About 
its  ends  as  centres  with  the  sides  as  radii  draw  circles  and 
then  complete  the  construction.  How  many  A  fulfilling  the 
conditions  are  possible  ?     How  are  they  related  ? 

F.    When  the  three  medials  are  given  (Fig.  105). 


Solution.  Remember  that  the  medials  trisect  each  other ; 
construct  the  A  OBC  according  to  (E),  and  draw  OA 
counter  to  OM  and  twice  as  long. 

151.  Problfem  X.  —  To  construct  an  angle  of  given  size 
and  subtended  by  a  given  tract. 

Data  :    O  the  given  angle,  AB  the  given  tract  (Fig.  106). 

Solution.  Construct  the  angle  BAD  of  given  size  (how?), 
draw  the  mid-normal  of  AB,  meeting  AD  at  P;  also  the 
normal  to  AD  at  A,  meeting  the  mid-normal  at  S.     About 


CONS  TR  UC  TIONS. 


131 


.S"  as  a  centre  with  radius  SA  draw  a  circle  ;  it  touches  AD 
at  A  (why?).  The  vertex  V  of  the  required  angle  may  be 
anywhere  on  the  arc  A  VB  or  on  its  symmetric  A  V^B  (why  ?) . 


Fig.  io6. 


152.    Problem  XI. 

ray. 


To  draw  a  circle  tangent  to  a  given 


Solution.  About  any  point  S  with  a  radius  equal  to  the 
distance  of  ^S  from  the  ray,  Z,  draw  a  circle ;  it  will  be  a 
circle  required  (why  ?) .  If  the  circle  must  touch  the  ray  L 
at  a  given  point  P,  then  S  must  be  taken  on  the  normal  to 
L  through  P  (why  ?) .  If,  besides,  the  circle  must  go  through 
a  given  point  Q,  then  ^  must  also  be  on  the  mid-normal  of 
PQ  (why  ?) .     Hence  the  construction. 


132  GEOMETRY. 

153.  Problem  XII.  —  To  draw  a  circle  touching  two  given 
rays. 

The  centre  may  be  anywhere  on  either  mid-ray  (why?). 
If  now  the  circle  is  to  touch  a  third  given  ray,  the  centre 
must  be  also  on  another  mid-ray ;  that  is,  it  must  be  the 
intersection  of  two  mid-rays  of  the  three  angles  of  the  three 
rays.  There  are  four  such  intersections  —  what  are  they? 
Complete  the  construction.     See  Fig.  60. 

154.  Problem  XIII.  —  To  draiv  a  circle  through  two 
points. 

The  centre  .S  may  be  anywhere  on  the  mid-normal  of  the 
tract  AB  between  the  points  (why?),  the  radius  is  —  what? 
If  now  the  circle  is  to  pass  through  a  third  point  C,  then  S 
must  also  be  on  the  mid-normal  oi  BC  and  CA  (why?). 
There  is  one,  and  only  one,  such  point  (why?)  ;  complete 
the  construction.     When  is  the  construction  impossible  ? 

155.  Problem  XIV.  —  To  draw  a  circle  through  two  given 
points  and  tangent  to  a  given  ray ;  or,  tangent  to  two  given 
rays  and  through  a  given  point. 

This  double  problem  is  mentioned  here  because  it  must 
naturally  present  itself  to  the  mind  of  the  student ;  but  the 
solution  involves  deeper  relations  than  we  have  yet  explored. 
See  Art.  000. 

Several  of  the  foregoing  problems  were  indefinite,  admit- 
ting any  number  of  solutions  :  these  latter  taken  all  together 
form  a  system  or  family.  Problems  concerning  parallelo- 
grams and  other  4-sides  may  often  be  solved  on  cutting  the 
4-side  into  two  A. 

156.  Problem  XV.  —  To  inscribe  a  regular  4-side  (square) 
in  a  circle  (Fig.  107). 


CONS  TR  UC  TIONS.  133 

Solution.  Join  consecutively  the  ends  of  two  conjugate 
diameters.  The  4-side  formed  is  inscribed  (why?)  and  is 
a  square  (why  ?) . 


Fig.  107. 

157.  Problem  XVI.  —  To  inscribe  a  regular  6-side  in  a 
circle. 

Solution.  Apply  the  radius  six  times  consecutively  as  a 
chord  to  the  circle  (Fig.  108).  The  figure  formed  will 
be  the  regular  6-side  (why?). 


Fig.  108. 


N.B.  This  seems  to  have  been  one  of  the  first  geometric 
problems  ever  solved.  The  Babylonians  discovered  that 
six  radii  thus  applied  would  compass  the  circle,  and  having 


134 


GEOMETRY. 


already  divided  the  circle  into  360  steps,  they  accordingly 
divided  this  number  by  6  and  thus  obtained  60  as  the 
basis  of  the  famous  sexagesimal  notation,  which  long  domi- 
nated mathematics  and  still  maintains  its  authority  un- 
diminished in  astronomy  aiid  chronometry. 

In  more  difficult  problems  it  is  often  advisable,  or  in  fact 
necessary,  to  suppose  the  problem  solved,  the  construction 
made,  and  investigate  the  relations  thus  brought  to  light. 
Then  the  facts  thus  discovered  may  be  used  regressively 
in  making  the  construction  required.  This  method  is  illus- 
trated in  the  following  :  ' 

158.  Problem  XVII.  —  To  draw  a  square  with  each  of 
its  sides  through  a  given  point. 

Let  A,  B,  C,  D,  be  the  four  given  points,  and  suppose 
(Fig.  109)  FQRS  to  be  the  square  properly  drawn.     Draw 


iO 


IQ 


G 

'^>< 

S 

/' 

F 

IG, 

109. 

AB,  cutting  a  side  of  the  square,  and  through  B  draw  BE 
parallel  to  the  side  cut.  Through  a  third  point  C  draw  a 
normal  to  AB,  meeting  QR  in  F.  Also  draw  FG  parallel 
to  FQ.    Then  the  A  ABE  and  CFG  are  congruent  (why  ?) . 


CONS  TR  UC  TIONS. 


135 


Hence  we  discover  that  CF—  AB.     This  fact  is  the  key  to 

the 

Solution.  Join  two  points  A  and  B ;  from  a  third,  C,  lay 
off  CF  equal  and  normal  to  AB.  The  join  of  D,  the  fourth 
point,  and  7^ is  one  side  of  the  square  in  position  (why?). 
Let  the  student  complete  the  construction  and  Show  that 
four  squares  are  possible. 

159.    Problem  XVIII.  —  To  trisect  a  given  angle. 

Suppose  the  problem  solved  and  the  ray  OT  to  make 
-^^TOB^^TOA  {Y\%.\\o).  /w 


Fig. 


From  any  point  A  of  the  one  end  of  the  angle  draw  a 
parallel  and  a  normal  to  the  other  end ;  also  draw  to  the 
trisector  a  tract  AS  =  OA.  Then  the  following  relations  are 
evident : 

-^AOS^-^fASO^^-^  SAT^^  STA) 

but  '^A0S=2'^.  TOB  =  2  STA  ; 

hence  '^STA  =  ^  SAT,  and  ST=  SA. 


136  GEOMETRY. 

Again,  2C  SAR  ■=  ^  SRA,  being  complements  of  equal 
angles  ;  hence  SA  —  SR,   TR  =  2  OA.     Hence 

Solution.  From  any  point  A  of  either  arm  of  the  given 
angle  draw  a  parallel  and  a  normal  to  the  other  arm ;  then, 
with  one  point  of  a  straight-edge  fixed  at  the  vertex  O,  turn 
the  edge  until  the  intercept  between  the  normal  and  the 
parallel  equals  2  OA.  But  to  do  this  we  need  a  graduated 
edge,  or  a  sUding  length  2  OA  on  the  edge  itself.  Accord- 
ingly, this  construction,  while  simple,  useful,  and  interesting, 
is  not  elementary  geometric  in  the  sense  already  defined. 
To  discover  such  a  solution  for  this  famous  problem,  has  up 
to  this  time  baffled  the  utmost  efforts  of  mathematicians. 

EXERCISES    II. 

1.  State  and  prove  the  reciprocals  of  Exercises  9,  10, 
II,  page  71. 

2.  Find  a  point  on  a  given  ray,  the  sum  of  whose  dis- 
tances from  two  fixed  points  is  a  minimum. 

3.  The  same  as  the  foregoing,  difference  supplacing 
sum. 

4.  A  and  A\  B  and  B\  C  and  C\  are  symmetric  as  to 
MN.     Show  that  AABC=A  A^B'C. 

5.  The  inner  and  outer  mid-rays  of  the  basal  angles  of 
a  symmetric  A  form  a  kite. 

6.  The  inner  mid-rays  of  the  angles  of  a  trapezium  form 
a  kite  with  two  right  angles. 

7.  The  joins,  of  the  mid-points  of  the  parallel  sides  of  an 
anti-parallelogram,  with  the  opposite  vertices,  form  a  kite. 

8.  The  mid-rays  of  the  angles  at  the  ends  of  the  trans- 
verse axis  of  a  kite  cut  the  sides  in  the  vertices  of  an  anti- 
parallelogram. 


EXERCISES  II.  137 

9.  How  must  a  billiard  ball  be  struck  so  as  to  rebound 
from  the  four  sides  of  a  table  and  return  through  its  original 
place  ? 

10.  Trace  a  ray  of  light  from  a  focus  F,  to  another  given 
point  Qf  reflected  from  a  convex  polygonal  mirror. 

11.  A  ray  of  light  falls  on  a  mirror  M,  is  reflected  along 
^  to  a  second  mirror  M\  is  thence  reflected  along  T.  Re- 
membering that  the  angle  of  incidence  equals  the  angle  of 
reflection,  show  that  the  angle  between  the  original  ray  R 
and  its  last  reflection  T  is  twice  the  angle  between  the 
mirrors  (angle  RT=  2  angle  MM^).  On  this  theorem  is 
grounded  the  use  of  the  sextant. 

12.  Two  mirrors  stand  on  a  plane  and  form  an  inner 
angle  of  60° ;  a  luminous  point  /'is  on  the  mid-ray  of  this 
angle  (or  anywhere  within  it)  ;  how  many  images  of  /'are 
formed?  How  are  they  placed?  What  if  the  angle  of  the 
mirrors  be  i/n  of  a  round  angle? 

This  theorem  is  beautifully  illustrated  in  the  kaleidoscope. 

13.  A  regular  /z-side  has  n  axes  of  symmetry  concurring 
in  the  centre  of  the  ;/-side,  which  centre  is  equidistant  from 
the  sides  of  the  //-side. 

14.  How  do  these  axes  lie  when  71  is  even?  when  n  is 
odd?  Show  that  if  n  be  even,  the  centre  is  a  centre  of 
symmetry. 

1 5 .  The  half-rays  from  centre  to  vertices  of  a  regular  «-side 
form  a  regular  pencil  of  n  half-rays,  and  those  from  the  cen- 
tre normal  to  the  sides,  another  regular  pencil ;  also  the  half- 
rays  of  each  pencil  bisect  the  angles  of  the  other. 

16.  In  a  figure  with  two  rectangular  axes  of  symmetry 
each  point,  with  three  others,  determines  a  rectangle,  and 
each  ray,  with  three  others,  a  rhombus. 


138  GEOMETRY, 

17.  Find  the  axes  of  symmetry  of  two  given  tracts. 

18.  A  regular  A,  along  with  the  figure  symmetric  with  it 
as  to  its  centre,  determines  a  regular  6-angle  (6-pointed 
star) . 

19.  Two  congruent  squares,  the  diagonals  of  one  lying  on 
the  mid-parallels  of  the  other,  form  a  regular  8-angle ;  also 
find  the  lengths  of  the  intercepts  at  the  corners. 

20.  The  outer  angle  of  a  regular  ?z-side  is  m  times  the 
outer  angle  of  a  regular  w/z-side. 

EXERCISES    III. 

1.  A  circle  with  its  centre  on  the  mid-ray  of  an  angle 
makes  equal  intercepts  on  its  arms. 

2.  Tangents  parallel  to  a  chord  bisect  the  subtended 
arcs,  and  conversely. 

3.  Tangents  at  the  end  of  a  diameter  are  parallel. 

4.  A  and  B  are  ends  of  a  diameter,  C  and  D  any  other 
two  points  of  a  circle ;  E  is  on  the  diameter,  and  angle 
AED  =  2  angle  CAD ;  find  the  possible  positions  of  E. 

5.  From  n  points  are  drawn  2  n  equal  tangent-lengths  to 
a  circle  ;  where  do  the  points  lie  ? 

6.  In  a  circumscribed  hexagon,  or  any  circumscribed 
2  «-side,  the  sums  of  the  alternate  sides  are  equal. 

7.  If  the  vertices  of  a  circumscribed  quadrangle,  hexagon, 
or  any  2  ;?-side,  be  joined  with  the  centre  of  the  circle,  the 
sums  of  the  alternate  central  angles  will  be  equal. 

8.  The  sums  of  the  alternate  angles  of  an  encyclic  2n- 
side  are  equal,  namely,  each  sum  is  {n  —  \)  straight  angles. 

9.  The  joins  of  the  ends  of  two  parallel  chords  are  sym- 
metric as  to  the  conjugate  diameter  of  the  chords. 


EXERCISES  III.  139 

10.  A  centre  ray  is  cut  by  two  parallel  tangents.  Show 
that  the  intercepts  between  tangent  and  circle  are  equal. 

11.  Normals  to  a  chord  from  the  ends  of  a  diameter 
make,  with  the  circle,  equal  intercepts  on  the  chord. 

12.  The  joins  of  the  ends  of  two  diameters  are  parallel  in 
pairs,  and  form  a  rectangle,  and  meet  any  two  parallel  tan- 
gents in  points  symmetric  in  pairs  as  to  the  centre. 

13.  The  joins  of  the  ends  of  two  parallel  chords  meet  the 
tangents  normal  to  the  chords  in  points  whose  other  joins 
are  parallel  to  the  chords. 

14.  A  chord  AB  is  prolonged  to  C,  making  BC  —  radius, 
and  the  centre  ray  CD  is  drawn  ;  show  that  one  intercepted 
arc  is  thrice  the  other. 

15.  The  intercepts,  on  a  secant,  of  two  concentric  circles 
are  equal. 

16.  A  chord  through  the  point  of  touch  of  two  tangent 
circles  subtends  equal  central  angles  in  the  circles. 

1 7.  Two  rays  through  the  point  of  touch  of  two  tangent 
circles  intercept  arcs  in  the  circles  whose  chords  are  parallel. 

18.  The  transverse  joins  of  the  ends  of  parallel  diameters 
in  two  tangent  circles  go  through  the  point  of  tangence. 

19.  Four  circles  touch  each  other  outerly  in  pairs  :  ist 
and  2d,  2d  and  3d,  3d  and  4th,  4th  and  ist ;  show  that  the 
points  of  touch  are  encyclic. 

20.  Show  that  three  circles  drawn  on  three  diameters  OA, 
OB,  OC  intersect  on  the  sides  of  the  A  ABC. 

21.  Find  the  shortest  and  the  longest  chord  through  a 
point  within  a  circle. 

22.  In  a  convex  4-side  the  sum  of  the  diagonals  is  greater 
than  the  sum  of  two  opposite  sides,  less  than  the  sum  of  all 
the  sides,  and  greater  than  half  the  sum  of  the  sides. 


140  GEOMETRY. 

23.  Three  half-rays  trisect  the  round  angle  O  \  on  each 
is  taken  any  point,  as  A,  B,  C.  Find  a  point  M  such  that 
the  sum  AfA  +  MB  +  MC  is  the  least  possible  (a  minimum). 

24.  Two  tangents  to  a  circle  meet  at  a  point  distant  twice 
the  radius,  from  the  centre ;  what  angles  do  they  form  ? 

25.  The  intercept  of  two  circles  on  a  ray  through  one  of 
their  common  points  subtends  a  constant  angle  at  the  other. 

26.  What  is  the  envelope  of  equal  chords  of  a  circle? 

27.  Two  movable  tangents  to  a  circle  intersect  under  con- 
stant angles ;  find  the  envelope  of  the  mid-rays  of  these 
angles. 

28.  The  vertex  Fof  a  revolving  right  angle  is  fixed  mid- 
way between  two  parallels,  and  its  arms  cut  the  parallels  at 
A  and  B ;  find  the  envelope  of  AB. 

29.  From  a  fixed  point  B  a  normal  BN  is  drawn  to  a 
movable  tangent  7"  of  a  circle,  and  through  the  mid-point  M 
of /W  there  is  drawn  a  parallel  to  T;  find  its  envelope. 

30.  The  vertices  of  a  A  are  Fj,  Fg,  Fg ;  the  mid-points  of 
its  sides  are  M^,  M,,,  J/i ;  the  feet  of  its  altitudes  are 
Ai,  A2,  As ;  the  inner  bisectors  of  its  angles  meet  the  oppo- 
site sides  at  B^,  B.,,  B^;  and  the  outer  bisectors  at 
B\,  B'2,  B\ ;  its  centroid  is  C,  its  in-centre  is  /,  its  circum- 
centre  is  S ;  its  angles  are  ai,  a^,  a^,  and  their  complements 
are  a\,  a'2,  a\.  Express  through  these  six  angles  the 
angles  between :  (i)  Fi^iandFiF;  (2)  A^A^Sind  Fz^s', 
(3)  AiAoamd  V-^A^;  (4)  A^Az  a.nd  A^A^ ;  (5)  MiA^  a.nd 
V^V,;  (6)  J/i^sand  K.V^;  (7)  J/j^s and  J/^^a ;  {d>)  A.M^ 
and  A^M^;  (9)  A^M.,  and  A^Ao^;  (10)  SV^  and  SV^] 
(11)  SV^  and  V,V,',  (12)  SV^  and  V^A^;  (13)  IV^  and 
JV\',   (14)  /Fiand  V^A.,;    (15)    F^'j  and  V^B'^. 


EXERCISES  IV.  141 

31.  Find  the  locus  of  the  mid-points  of  chords  through  a 
fixed  point  upon,  within,  or  without  a  fixed  circle. 

32.  Find  the  locus  of  the  mid-points  of  the  intercepts  of 
a  secant  between  a  fixed  point  and  a  fixed  circle. 

33.  As  the  ends  of  a  ruler  slide  along  two  grooves  normal 
to  each  other,  how  does  its  mid-point  move  ? 

34.  Two  equal  hoops  move  along  grooves  normal  to  each 
other  and  touch  each  other ;  how  does  the  point  of  touch 
move? 

35.  Orthocentre  O,  centroid  C,  circum-centre  6",  and  cen- 
tre F  of  Feuerbach's  (9-point)  circle,  of  a  A  are  collinear, 
and  0C=2CS  (Euler),  OF=FS. 

36.  Two  parallel  tangents  meet  two  diameters  of  a  circle 
at  the  vertices  of  a  parallelogram  concentric  with  the  circle. 

37.  The  inner  mid-rays  of  the  angles  of  a  4-side  form  an 
encyclic  4-side. 

38.  The  outer  mid-rays  of  the  angles  of  a  4-side  form  an 
encyclic  4-side.     How  are  the  4-sides  of  37  and  2>^  related? 

39.  The  circum-centres  of  the  four  A  into  which  a  4-side 
is  cut  by  its  diagonals  are  the  vertices  of  a  parallelogram. 

40.  The  circum-centres  of  the  two  pairs  of  A,  into  which 
a  4-side  is  cut  by  its  diagonals  in  turn,  are  how  related  to 
each  other  and  to  the  centres  in  39  ? 

EXERCISES    IV. 

1.  Construct  a  square,  knowing 
{a)  Its  side ;  or  (6)  its  diagonal. 

2.  Construct  a  rectangle,  knowing 

{a)  Two  sides ;  or  {d)  a  side  and  a  diagonal ;  or  {c) 
either  a  side  or  a  diagonal  and  the  angle  of  either  with  the 
other ;  or  {d)  a  diagonal  and  its  angles  with  the  other 
diagonal. 


142  GEOMETRY. 

3.  Construct  a  parallelogram,  knowing 

{a)  Two  sides  and  one  angle ;  or  {b^  a  side,  a  diagonal, 
and  the  included  angle ;  or  (r)  two  sides  and  the  opposite 
diagonal ;  or  (^)  two  sides  and  the  included  diagonal ;  or 
(<f)  two  diagonals  and  a  side ;  or  (/)  two  diagonals  and 
their  angles  with  each  other. 

4.  Construct  an  anti-parallelogram,  knowing 

{a)  Its  parallel  sides  and  the  distance  between  them ; 
(J))  two  adjacent  sides  and  their  included  angle ;  {c)  two 
adjacent  sides  and  the  angle  between  the  non-parallel  sides ; 
(^)  a  diagonal  and  two  adjacent  sides ;  (<?)  a  diagonal,  a 
side,  and  the  included  angle. 

5.  Construct  a  kite,  knowing 

(^a)  Two  sides  and  an  axis;  {b^  two  sides  and  the  in- 
cluded angle  ;   {c)  a  side  and  the  axes. 

6.  Construct    the    rays    equidistant    from    three    given 
points. 

7.  Draw  a  ray  through  a  given  point  equally  sloped  to 
two  given  rays. 

8.  A  square  has  one  vertex  at  a  given  point,  and  two 
others  on  two  given  parallel  rays ;  draw  it. 

9.  Hypotenuse  and  sum  of  sides  of  a  right  A  are  given  ; 
draw  it. 

10.  Construct  a  regular  2"- side,  and  a  regular  3.2'*-side. 

11.  Find  the  centre  of  a  given  circular  arc. 

12.  Trisect  a  right  angle. 

13.  Two  points,  A  and  B,  of  a  ray  are  given;  find  any 
number  of  points  of  the  ray  without  drawing  it,  and  without 
opening  the  compasses  more  than  AB. 

14.  Find  a  point  on  a  given  ray  or  given  circle  that  has 
a  given  tangent-length  with  respect  to  a  given  circle. 

15.  Through  a  given  point  draw  a  secant  on  which  a 
given  circle  shall  make  a  given  intercept. 


EXERCISES  IV.  143 

1 6.  Draw  four  common  tangents  to  two  given  circles. 

17.  Draw  a  ray  touching  a  given  circle  and  equidistant 
from  two  given  points. 

18.  Draw  a  ray  on  which  two  given  circles  shall  make 
two  given  intercepts. 

19.  With  three  given  radii  draw  three  circles,  each  touch- 
ing the  other  two. 

20.  Draw  a  circle  touching  the  radii  and  the  arc  of  a 
given  sector. 

21.  Draw  a  circle  touching  two  given  equal  intersecting 
circles  and  their  centre  ray. 

22.  On  the  central  intercept  of  two  equal  intersecting 
circles  as  diameter  draw  a  circle  ;  then  draw  a  circle  touch- 
ing the  three  circles. 

23.  Three  equal  circles  touch  each  other  outerly ;  draw 
a  circle  touching  the  three. 

24.  Find  a  point  from  which  two  given  apposed  tracts 
appear  to  be  equal. 

25.  Through  two  given  points  of  a  given  circle  draw  a 
circle  that  shall  cut  a  third  circle  orthogonally. 

26.  Construct  a  A,  knowing 

{a)  The  feet  of  the  altitua  .s  ;  {b^  the  foot  of  one  altitude 
and  the  mid-points  of  the  other  two  sides ;  (r)  the  three 
ex-centres  ;   {d)  two  ex-centres  and  the  in-centre. 

27.  Draw  through  a  given  point  a  ray  that  shall  form  with 
the  sides  of  a  given  angle  a  A  of  given  perimeter.  Hint : 
Use  the  properties  of  ex-circles. 

28.  Draw  a  5 -side,  knowing  the  mid-points  of  the  sides. 

29.  On  a  tract  AB  there  is  drawn  a  regular  3-side ;  draw 
on  it  a  regular  6-side.  Generalize  the  problem,  changing 
3  into  n,  6  into  211,  and  solve  it. 

30.  Given  a  regular  ;z-side  ;  draw  a  regular  2;2-side  having 
the  original  //  vertices  for  alternate  vertices.  Do  not  use 
the  circumcircle. 


144  GEOMETRY. 

AREAL    RELATIONS. 

Geometrica  geometrice. 

1 60.  Hitherto  our  attention  has  been  fastened  exclusively 
on  points  and  lines  as  composing  figures.  We  have  regarded 
no  higher  extents  than  those  of  one  dimension.  The  surface, 
or  extent  of  two  dimensions,  bounded  by  a  circle  or  the  sides 
of  a  triangle,  we  have  not  considered  at  all,  but  only  the 
circle  or  triangle  itself.  Moreover,  our  comparisons  as  to 
size  have  referred  exclusively  to  magnitudes  directly  super- 
posable  and  homoeoidal  both  in  themselves  and  between 
themselves,  such  as  tracts,  angles,  arcs  of  the  same  circle. 
Such  comparisons  are  not  suited  to  set  forth  the  sharp  dis- 
tinction between  congruence  and  equality,  inasmuch  as  con- 
gruent tracts,  angles,  arcs  are  equal,  and  equal  tracts,  angles, 
arcs  are  congruent.  But,  as  we  now  advance  to  the  discus- 
sion of  two-dimensional  extents,  the  distinction  in  question 
becomes  essential  and  regulative.  Accordingly  we  premise 
the  following : 

Def.  A  surface  or  two-dimensional  extent  considered 
solely  as  to  its  size,  or  the  amount  of  a  two-dimensional 
extent,  is  called  an  Area. 

N.B.  Where  no  confusion  will  result,  we  shall  designate 
the  area  bounded  by  a  border  by  the  name  of  the  border 
itself :  thus,  circle  for  the  area  bounded  by  a  circle,  etc. 

161.  As  we  shall  have  frequent  occasion  to  compound 
two  areas  into  one  and  to  divide  one  area  into  two,  in  order 
to  treat  these  processes  logically  we  must  first  define  them 
precisely. 

Def.  If  any  two  points  of  the  border  of  a  surface  bounded 
by  a  single  continuous  line  be  joined,  the  surface  is  said  to 


AREAL  RELATIONS. 


145 


be  cut  or  divided  into  two  parts,  and  the  section-line  counts 
as  part-border  both  of  the  one  part  and  of  the  other.  Thus 
SL  is  such  a  section-line  (Fig.  in). 


Fig.  hi. 

N.B.  It  is  essential  that  the  border  be  single  and  con- 
tinuous ;  that  is,  that  the  surface  be  simply  compendent.  If 
the  surface  be  doubly  compendent,  as  a  ring,  then  the 
section-line  may  or  may  not  cut  it  into  two  parts.  This 
doctrine  of  the  compendency  of  surfaces  is  a  creation  of 
Riemann's,  with  which  we  have  at  present  no  further  con- 
cern (Fig.  112). 


Fig.  112. 


Any  part  of  the  border  of  a  surface  may  be  treated  as  the 
beginning  or  the  end  of  the  surface,  and  all  the  rest  of  the 
border  as  the  end  or  the  beginning. 


146 


GEOMETRY. 


162.  When  two  areas,  or  surfaces,  are  apposed,  the 
beginning  of  the  second  fitting  on  the  end  of  the  first,  the 
two  congruent  part-borders  herewith  ceasing  to  be  any  part 
of  the  border  of  either,  the  whole  area  bounded  by  the 
remaining  part-borders  of  the  two,  namely,  the  beginning  of 
the  first  and  the  end  of- the  second,  is  called  the  sum  of  the 
two  areas  thus  apposed  ;  e.g.  we  may  appose  two  equal  semi- 
circles and  get  a  circle  as  the  sum  ;  or  two  congruent  A  and 


Fig.  113. 

get  a  parallelogram  as  the  sum  ;  or  two  congruent  right  A 
and  get  a  symmetric  A  as  sum ;  or  two  symmetric  A  and 
get  a  kite  as  sum ;  or  a  parallelogram  and  a  symmetric  A 
and  get  an  anti-parallelogram  as  sum  (Fig.  113). 

Def.    The  areas  apposed  are  called  parts  or  summands  of 
the  sum. 


CRITERIA    OF  EQUALITY.  147 

CRITERIA   OF    EQUALITY. 

163.  I.  Two  surfaces  not  congruent  but  divisible  into  the 
same  number  of  parts  so  that  for  each  part  of  either  there  is 
a  congruent  part  of  the  other,  are  said  to  be  equal  in  area, 
to  agree  in  area,  to  have  equal  areas,  or  simply  to  be  equal. 
These  four  phrases  may  be  used  indifferently,  according  to 
convenience. 

2.  Two  surfaces  not  congruent,  but  which  may  be  made 
congruent  by  addition,  subtraction,  or  both,  in  case  of  each, 
of  the  same  surfaces,  or  surfaces  congruent  in  pairs,  are  said 
to  be  equal  in  area. 

3.  Two  surfaces,  each  equal  by  either  of  the  foregoing 
tests,  to  the  same  third  surface,  or  to  one  of  two  equal  sur- 
faces, are  themselves  said  to  be  equal. 

These  three  criteria  will  serve  our  present  purposes.  A 
most  important  fourth  criterion  will  be  introduced  at  the 
proper  place. 

It  thus  appears  that  while  congruence  implies  sameness 
as  to  both  size  and  shape,  equality  implies  sameness  as  to  size 
only. 

164.  When  the  borders  of  two  plane  surfaces  (the  only 
ones  that  we  deal  with)  are  congruent,  the  surfaces  are 
themselves  congruent  and  their  areas  are  equal.  This  fol- 
lows from  the  homoeoidality  of  the  plane.  For  let  A  and  A\ 
B  and  B\  be  two  pairs  of  corresponding  points  in  the 
borders  ;  then  the  ray  AB  will  fit  on  the  ray  AB,  point  for 
point  throughout ;  and  so  for  any  other  pair  of  correspond- 
ing rays.  Thus  the  one  surface  fits  point  for  point,  line  for 
line,  on  the  other. 


148 


GEOMETRY. 


[Th.  LXXX. 


165.  The  conditions  of  congruence  among  A  are  known. 
The  most  important  condition  of  congruence  between  two 
parallelograms  is  this  : 

Theorem  LXXX.  —  Two  parallelograms  with  the  sides  of 
the  o?ie  equal  respectively  to  the  sides  of  the  other,  and  the 
angles  of  the  one  either  equal  or  supplemental  respectively  to 
the  angles  of  the  other,  are  congruent  (Fig.  114). 


Fig.  114. 

For,  \{  a  =^  a\  h  —  b\  and  a  =  a',  then  plainly  we  may- 
fit  the  one  parallelogram  perfectly  on  the  other.  But  if  a  = 
supplement  of  a',  then  a  =  fi' ,  and  we  may  again  fit  the 
parallelograms.     Q.  e.  d. 

Corollary.  Two  rectangles  having  a  pair  of  adjacent  sides 
of  one  equal  to  a  pair  of  adjacent  sides  of  the  other  are 
congruent. 


166.  Def  Any  tract  forming  part  of  the  border  of  a 
closed  figure  and  treated  as  its  lowest  part  may  be  called  its 
base. 

Def.  The  greatest  normal  distance  from  a  point  of  the 
figure  to  this  base  ray  is  called  the  altitude  or  height  of  the 
figure  (Fig.  115). 

In  the  diagrams  a  denotes  altitude  and  b  base  —  a  con- 
venient notation,  which  will  be  generally  employed. 

The  notions  of  base  and  altitude  are  correlate,  implying 


Th.  LXXXL]        criteria    of  equality.  149 

each  other,  but  they  are  not  in  general  interchangeable  ;  the 
relation  between  them  is  not  a  mutual  or  reciprocal  relation. 

Def.   The  base  and  altitude  of  an  areal  figure  may  be 
called  its  two  dimensions. 


167.  In  only  one  figure  are  these  dimensions  interchange- 
able, namely,  in  the  rectangle,  and  it  is  this  fact  that  recom- 
mends the  rectangle  as  the  standard  figure  in  comparison  of 
areas. 

A  rectangle  whose  two  adjacent  sides,  or  whose  altitude 
and  base,  or  whose  two  dimensions,  are  a  and  b,  will  be 
denoted  by  the  symbol  rect.  ab,  or  I  I  ab,  or  simply  ab, 
rectangle  being  always  understood. 

\i  a  =  AB  and  b  =  BC,^t  may  write  AB-BC  and  read 
rectaftgle  of  AB  and  BC,  the  dot  (•)  being  used  not  to 
denote  multipUcation  but  merely  to  separate  and  distinguish 
base  and  altitude. 

1 68.  Theorem  LXXXI.  —  The  sum  of  two  rectangles  that 
agree  in  one  dimension  is  a  third  rectangle  agreeing  ivith  each 
in  that  di^nension  and  having  for  its  second  dimension  the 
sum  of  the  second  dimensions  of  the  summands. 

Data:  ABCD  and  EFGH  two  rectangles  agreeing  in 
one  dimension,  BC  =  EH. 

Proof.  Appose  BC  and  EH  ;  then  AF  becomes  a  single 
tract  (why?),  namely,  the  sum  of  the  tracts  AB  and  EF. 


150 


GEOMETRY. 


[Th.  LXXXII. 


Also  DG  becomes  similarly  a  single  tract  equal  to  AF. 
Hence  AFGD  is  a  rectangle  (why?),  and  by  definition  it 
is  the  sum  of  the  two  component  rectangles ;  moreover,  it 


c 

a 

b 

a 

A  BE  F 

Fig.  ii6. 

agrees  with  each  in  one  dimension,  while  its  second  dimen- 
sion, AF^  is  the  sum  of  the  two  second  dimensions,  AB  and 

EF.       Q.  E.  D. 

Corollary.     Symbolically 

ab-^a'b=  {a-\-a})b, 
aa'  -\-ab'  -\-ba' +bb'  =  a{a'  +  b')  +b{a^  -^b^)  =  {a-\-b){a' +b') . 
Draw  a  figure  exhibiting  this  last  relation. 

169.  Theorem  LXXXII.  —  Two  rectangles  that  agree  in 
both  dimensions  are  equal. 

For  plainly  they  are  congruent,     q.  e.  d. 

170.  Theorem  LXXXIII.  —  Two  rectangles  agreeing  in  one 
dimension  but  not  in  the  other  are  unequal. 

Data:    A  BCD  and  EFGH  two  rectangles  agreeing  in 
one  dimension,  AB  =  FF,  but  not  in  the  other,  BC=^  EH. 

Proof.    Fit  AB  on  EF;     then  DC  will  fall  above  or 
below  HG  according  as  BC>EH  or  BC<EB;     the 


th.  lxxxv.]    criteria  of  equality. 


151 


whole  of  one  rectangle  fits  on  part  of  the  other,  i.e.  they 

are  unequal,  q.  e.  d. 

Corollary.  The  rectangle  with  the  greater  dimension  is 
the  greater. 


E  FA  B 

Fig.  117. 

171.  Theorem  LXXXIV.  —  Two  rectangles  agreeing  in 
area  and  in  one  dimension  agree  in  the  other  also. 

Proof.  For  if  unequal  in  the  second  dimension  they  would 
be  unequal  in  area  (why  ?)  ;  but  they  are  equal  in  area ; 
hence,  etc.     q.e.  d. 

N.B.    In  symbols, 

if  ab  =  cd  and   a  =  c,   then   b  =^  d. 

172.  Theorem  LXXXV. — A  rectangle  and  a  parallelo- 
gram that  agree  in  both  dimensions  agree  in  area  also. 


Data:    A  rectangle  ABCD  and  a  parallelogram  EFGH, 
having  AB  =  EF  and  AD  =  HH\ 


152  GEOMETRY,  [Th.  LXXXVI. 

Proof.  Draw  HH^  and  GG'  normal  to  EF.  Then  the 
right  A  EH^H  and  FG^G  are  congruent  (why?)  ;  take 
away  the  first  from  the  trapezoid  EG^GH,  and  there  is 
left  the  rectangle  B'G^GH;  take  away  the  second,  and 
there  is  left  the  parallelogram  EFGH ;  hence  the  paral- 
lelogram equals  the  rectangle.  Moreover,  this  latter  is 
congruent  with  the  rectangle  A  BCD  (why?)  ;  hence  the 
parallelogram  EFGH  tq}X2i\s  the  rectangle  ABCD.    q.e.d. 

Corollary  i.  Parallelograms  agreeing  in  both  dimensions 
are  equal. 

Corollary  2.  Parallelograms  agreeing  in  area  and  in  one 
dimension  agree  in  the  other. 

Corollary  3.  Parallelograms  agreeing  in  one  but  not  in 
the  other  dimension  are  unequal. 

Corollary  4.  Parallelograms  agreeing  in  one  dimension 
but  not  in  area  do  not  agree  in  the  other  dimension. 

Corollary  5.  Parallelograms  agreeing  in  area  but  not  in 
one  dimension  do  not  agree  in  the  other. 

Corollary  6.  Equal  parallelograms  with  equal  bases  along 
the  same  ray,  and  lying  on  the  same  side  of  that  ray,  have 
the  sides  opposite  the  bases  in  the  same  ray. 

173.  Theorem  LXXXVI.  —  A  A  has  half  the  area  of  a 
recta7igle  with  the  same  dimensions. 

Proof.  Complete  the  A  ABC  into  the  parallelogram 
ABCD ;  then  ABC  and  DCB  are  two  congruent  A  whose 
sum  is  the  parallelogram  ABCD  of  the  same  dimensions ; 
i.e.  each  is  half  the  parallelogram ;  and  this  latter  has  the 
same  area  as  the  rectangle  of  the  same  dimensions ;  hence 
the  A  has  half  the  area  of  the  rectangle  of  the  same  dimen- 
sions.    Q.E.D. 


Th.  LXXXVL]     criteria    of  equality.  153 

Scholiu7n.  We  may  express  this  fact  by  saying  that  the 
A  equals  half  the  rectangle  (or  parallelogram)  of  its  base 
and  altitude. 

Corollayy  i.  A  A  is  determined  in  area  by  its  two  dimen- 
sions, base  and  altitude. 

Corollary  2.    A  agreeing  in  dimensions  agree  in  area. 

Corollary  3.  A  agreeing  in  area  and  in  one  dimension 
agree  in  the  other  also. 

Corollary  4.  A  with  equal  bases  along  the  same  ray  and 
with  vertices  in  either  of  two  parallels  equidistant  from  the 
base  are  equal. 

Corollary  5.  Equal  A  with  equal  bases  along  the  same 
ray  have  their  vertices  in  two  parallels  equidistant  from  the 
base. 

Corollary  6.     A  medial  bisects  the  area  of  the  A. 

Corollary  7.  In  general,  when  any  two  of  the  three  re- 
lated magnitudes,  base,  altitude,  and  area  of  rectangle,  par- 
allelogram or  A  are  known,  the  third  is  also  known  univo- 
cally. 

Corollary  8.  So  far  as  size  is  concerned,  the  dimensions 
in  rectangle,  parallelogram,  and  A  may  be  exchanged. 

174.  Since  a  A  is  half  the  rectangle  of  the  same  dimen- 
sions, and  since  we  can  add  rectangles  agreeing  in  one 
dimension,  we  may  also  add  A  agreeing  in  one  dimension, 
preserving  the  same.  Thus  by  apposing  the  bases  b  and  // 
of  two  (Fig.  119)  A  agreeing  in  altitude  a,  we  get  half  of  a 
rectangle  a  {b  -\-  b^),  equal  to  a  A  with  same  altitude  and 
base  b  -f  //,  the  sum  of  the  bases.  By  apposing  two  A 
along   the  common  base   b  we  get  half  of  the   rectangle 


154 


GEOMETRY. 


[Th.  LXXXVII. 


{a-\-a')  b  equal  to  a  triangle  with  the  common  base  b  and 
with  altitude  the  sum  of  the  altitudes.     Hence, 

Theorem  LXXXVII.  —  The  sum  of  two  A  agreeing  in  one 
dimension  equals  a  third  A  7vith  the  same  dimension,  its 
second  dimension  being  the  sum  of  the  second  dimensions  of 
the  summands. 

Scholium.    In  symbols 

ab  +  ab'  =  a{b^  ^'),  ab  +  a^b  =  {a  +  a^)  b. 


a 

19. 

/\ 

/\K 

/r\ 

\  ^ 

/ 

h 

Fig.  I 

MISCELLANEOUS    APPLICATIONS. 

*i75.  Theorem  LXXXVIII.  — /^7^^«  tivo  A  lie  apposed 
on  the  same  base:  I.  That  base  bisects  the  join  of  the  two 
vertices^  if  the  A  are  equal. 

Data:    ABC  and  ABC  the  equal  apposed  A  (Fig.  120). 

Proof.  The  altitudes  CD  and  CD^  are  equal  when  the 
A  are  equal  (why?)  ;  hence  A  CDI  and  CD' I  are  con- 
gruent;    hence  CI=  CI.     q.e.d. 

Conversely, 

II.  The  A  are  equal,  if  that  base  bisects  the  Join  of  the 
vertices. 


th.  lxxxix.]    miscellaneous  APPLLCATLONS.       155 

Proof.  The  A  CDI  and  CD^I  are  again  congruent 
(why?)  ;  hence  CD  =  CD\  hence  ABC  and  ABC  are 
equal,     q.  e.d. 

C 


Fig.  I20. 

Def.  Two  parallels  to  the  sides  of  a  parallelogram  through 
any  point  within  it  divide  it  into  four  parallelograms,  and 
each  pair  of  opposites  we  may  call  complemental. 

*i76.  Theorem  LXXXIX.  —  Whe7i  the  common  vertex 
of  the  complementals  is  on  the  diagonal,  the  paii'  on  opposite 
sides  of  the  diagonal  are  equal. 

Data:  ABCD  the  parallelogram,  P  the  point  on  the 
diagonal  AC,  PD  and  PB  the  complementals  (Fig.  121). 


156 


GEOMETRY. 


[Th.  XC. 


Proof.  From  the  equal  A  ADC  and  ABC  take  away  the 
pairs  of  equal  A  APH  and  APE,  CPG  and  CPF;  there 
remain  the  equal  complementals  PD  and  PB.     q.e.d. 

Conversely,  When  the  compleitienials  are  equal,  the  com- 
mon vertex  is  on  the  diagonal. 

Proof.  If  P  moves  off  from  the  diagonal,  as  towards  E, 
then  PG  will  be  lengthened  and  PE  shortened  without 
affecting  either  PH  or  PF;  i.e.  one  of  the  equal  comple- 
mentals will  be  increased  and  the  other  decreased  ;  hence, 
for  P  not  on  the  diagonal  the  complementals  are  unequal ; 
hence  the  theorem,  by  contraposition,     q.e.d. 

Corollary.  Parallelogram  CH  =  parallelogram  CE,  and 
parallelogram  AF  =  parallelogram  A  G. 

177.  Theorem  XC.  —  A  trapezoid  equals  half  the  rectangle 
of  its  altitude  and  the  sum  of  its  bases,  or  equals  the  rectangle 
of  its  altitude  a7id  the  mid-parallel  to  its  bases. 

Data:  A  BCD  a  trapezoid,  MP  the  mid-parallel  (Fig. 
122). 

Proof.    Draw  B^ C  parallel  to  AD]  hence,  etc.     q.e.d. 


r 


Fig.  122,  Fig.  123. 

178.    Theorem   XCI. — A  kite  equals  half  the  rectangle  of 
its  diagonals  (Fig.  123). 

Let  the  student  deduce  the  proof  from  the  figure. 


Th.  XCIL] 


SQUARES. 


157 


SQUARES. 

179.  We  have  seen  that  the  rectangle  is^  distinguished 
among  parallelograms  by  the  interchangeability  of  its  dimen- 
sions ;  among  rectangles  the  square  is  distinguished  by  the 
equality  of  its  dimensions.  Hence  it  is  determined  com- 
pletely by  one  dimension ;  hence  we  may  reason  about 
squares  and  compare  them  through  their  dimensions  more 
readily  than  is  possible  with  rectangles. 

180.  Theorem  XCII.  —  The  square  on  the  sum  of  two 
tracts  equals  the  sum  of  the  squares  on  the  tracts  and  twice 
the  rectangle  of  the  tracts. 

Data:  a  and  b  two  tracts,  AB  their  sum,  ABCD  the 
square  on  that  sum  (Fig.  124). 


a 

b 

P 

a 
a 

b 

A  E        B 

Fig.  124. 


Proof.  Draw  £G  parallel  to  BC  and  NF  parallel  to  CZ> ; 
they  cut  the  whole  square  into  four  parts,  namely,  the  square 
on  a,  the  square  on  l>,  and  the  two  congruent  rectangles  ad 
and  ab.     q.  e.  d. 

Corollary.  The  square  on  a  tract  equals  four  times  the 
square  on  half  the  tract. 


158  GEOMETRY.  [Th.  XCIII. 

i8i.  Theorem  XCIII.  —  The  square  on  the  difference  of 
two  tracts  equals  the  sum  of  the  squares  on  the  tracts,  less 
twice  the  rectangle  of  the  tracts. 

Proof.  In  the  preceding  figure  treat  AB  ox  a  -\-  b  z.%  the 
one  tract  and  EB  or  b  as  the  other  \  then  AE  or  a  is  the 
difference.  The  square  ^  C  is  the  square  on  the  tract  a  -{- b  \ 
increase  it  by  the  square  PC  on  b,  so  that  this  square  is  to  be 
counted  twice  and  thought  as  doubly  laid  in  the  figure  ;  now 
strip  off  the  rectangle  HC  or  (^  +  ^)  b,  and  its  congruent 
BG\  there  remains  the  square  AP  on  the  difference  a. 

Q.  E.  D. 

182.  Theorem  XCIV.  —  The  rectangle  of  the  sum  and  dif- 
ference of  two  tracts  equals  the  difference  of  the  squares  on 
those  tracts. 

Proof.  In  the  same  figure  treat  (^  +  <^  as  the  one  tract 
and  a  as  the  other,  so  that  2  ^  +  /^  is  the  sum  and  b  the  dif- 
ference of  the  tracts.  Then  the  two  rectangles  ^Cand  HG 
agree  in  one  dimension  b  and  the  sum  of  their  other  dimen- 
sions is  2  ^  -f-  /; ;  hence  their  sum  is  the  rectangle  {2  a  -{-  b')b ; 
i.e.  the  rectangle  of  the  sum  and  difference  of  the  tracts 
a-\-b  and  a.  Moreover,  this  area  is  plainly  what  is  left  on 
taking  away  the  square  on  a  from  the  square  on  {a-^  b)  -, 
hence,  etc.     q.e.d. 

Scholium.  Calling  the  tracts  u  and  v,  we  may  express 
these  theorems  in  symbols  thus  :  (//  -f  z')-  =//  -f  ^^"  +  2  uv ; 
{u  —  vY  =  u^  -{-7>^  —  2  uv  ;   {u  4- 1!)  {u  —  v)  =  u^  —  vK 

But  let  the  student  carefully  beware  of  importing  any 
algebraic  meaning  into  these  symbols  at  this  stage  of  the 
discussion  ;  u'^,  for  example,  does  not  mean  the  product  of  u 
multiplied  by  u,  but  merely  the  square  whose  side  is  //. 


Tk.  XCV.] 


SQUARES. 


159 


183.  Theorem  XCV.  —  The  square  on  the  hypotenuse  of 
a  right  A  equals  the  sum  of  the  squares  on  its  other  sides 
(Fig.  125). 


h     C 


h    C 


i  ^  / 

i    / 

1  ,--"'"  \ 

\ 

A' 
Fig.  125. 


Proof.  Let  AC  and  A^C  be  two  congruent  squares  on 
the  tract  a -\-  b  -,  take  away  from  each  the  four  congruent 
right  A,  I,  2,  3,  4 ;  there  is  left  of  the  one  the  square  on 
the  hypotenuse  of  the  right  A,  and  of  the  other  the  sum  of 
the  squares  on  the  legs,  a  and  b.     Q.  e.  d. 

Scholium.  This  most  famous  theorem  was  discovered  by 
Pythagoras  (circa  B.C.  550),  it  is  said,  while  pursuing  cer- 
tain arithmetical  researches.  He  was,  in  fact,  seeking  out 
pairs  of  numbers,  the  sum  of  whose  squares  was  itself  the 
square  of  a  number,  when  he  made  the  astonishing  observa- 
tion that  all  such  pairs,  when  measured  off  in  terms  of  a  unit 
length,  formed  the  two  legs  of  a  right  A  of  which  the  third 
number,  similarly  laid  off,  formed  the  hypotenuse.  Proofs 
of  the  proposition  abound.  In  the  classic  one  of  Euclid,  it 
is  shown  that  the  square  on  ^C=  rectangle  AD  (Fig.  126), 
the  media  of  comparison  being  the  congruent  A  BAE  and 
FAC.     Similarly  the  square  on  ^C=  rectangle -5Z>.     The 


160 


GEOMETRY. 


[Th.  XCV. 


demonstration  in  the  text  seems  to  be  the  most  simple  and 
direct  that  is  possible,  while  its  presuppositions  are  the  least 
possible. 


Fig.  126. 


184.  Naturally  we  now  inquire  into  the  relations  among 
the  squares  on  the  sides  of  oblique  A. 

Def.  The  foot  of  the  normal,  from  a  point  to  a  ray,  is 
called  the  (orthogonal)  projection  of  the  point  on  the  ray ; 
and  the  tract  between  the  projections  of  the  ends  of  a  tract 
is  called  the  projection  of  the  tract  itself  (Fig.  127). 


Thus  /"  and  Q  are  the  projections  of /'and  Q  on  Z,  and 
P^Q  is  the  projection  oi  PQ  on  L. 


Th.  XCVL] 


SQUARES. 


161 


185.  Theorem  XCVI.  —  The  square  on  any  side  of  a  tri- 
angle equals  the  sum  of  the  squares  on  the  other  sides,  in- 
creased or  decreased  by  tivice  the  rectangle  of  either  and  the 
projection  of  the  other  upon  it,  according  as  the  first  side 
lies  opposite  an  obtuse  or  an  acute  angle. 

Data:  ABC  the  A,  BD  the  projection  of  BC  on  AB 
(Fig.  128). 

C 


Proof. 

and 
hence 
But 
hence 


AC^AEC^  CD\ 

CD'  =BC--  BD-  (why  ?)  ; 

A  C-  =  AD-  -  BD'-  +  BC\ 

AD-  -  BD  =  AB  {AB  +  2  BD)  (why  ?)  ; 

^  C^  =  AB^  +BC'^2  AB'BD. 


Thus  far  ji  has  been  considered  obtuse ;  if  it  be  acute, 
then  AD=  AB  +  BD  ;  but  BD  is  to  be  reckoned  leftward, 
oppositely  to  BD  in  the  former  case  ;  that  is,  BD  is  reversed 
in  sense,  a  fact  which  we  may  express  in  symbols  by  writing 

AD  =  AB-  DB. 

Hence  results,  as  before, 

AC^  =  AB-  +  BC-  ~2AB'  DB.  q.e.d. 


162  GEOMETRY.  [Th.  XCVII. 

Scholium.   We  observe  that  when  /8  is  a  right  angle,  then, 
and  then  only,  D  falls  on  B,  the  projection  on  BD  vanishes, 
and  there  results  the  Pythagorean  Theorem, 
AC^AB'^^BCK 

As  /3  changes  from  obtuse  to  acute,  the  point  B  passes 
from  the  left  to  the  right  of  D,  and  the  tract  BD  changes 
its  sense,  —  from  being  reckoned  rightv^dsdi  it  comes  to  be 
reckoned  left^'dxA.  It  is  this  change  of  sense  in  BD  that 
changes  the  addition  into  the  subtraction  of  the  rectangle. 
It  is  often  absolutely  necessary  to  take  account  of  the  sense 
of  a  magnitude,  the  way  it  is  reckoned,  in  order  to  perceive 
the  generality,  the  internal  coherence  and  continuity,  of  our 
results. 

Corollary.  When  the  square  on  one  side  of  a  A  equals 
the  sum  of  the  squares  on  the  others,  the  A  is  right-angled 
opposite  that  side.     Converse  of  the  Pythagorean  Theorem. 

1 86.  Theorem  XCVII.  The  sum  of  the  squares  on  two 
sides  of  a  A  equals  twice  the  sum  of  the  squares  oti  half 
the  third  side  and  its  medial. 

Data:  ABC  the  A,  CM  medial,  and  CN  normal  to 
AB  (Fig.  129). 


Proof.       AC''  =  am''  +  CaP -f  2  AM-  MN, 
BC'  =  ¥M^  +  CM''-2BM'MN. 


Th.  XCVIII.] 


SQUARES. 


163 


Adding  and  remembering  that  AM=  BM,  we  get 


AC  \BC  =2{AM  -{-  CM). 


Q.  E.  D. 


Corollary  i.    If  a,  b,  c  be  the  sides  of  the  A,  and  tn  the 
medial  of  c,  then 


4;;r 


2a^ 


2  b' 


Corollary  2.  If  the  A  be  regular,  then  ;//  is  the  altitude, 
and  if  J-  be  a  side, 

*i87.  Theorem  XCVIII.  — The  sum  of  the  squares  on  the 
sides  of  a  quadrangle  equals  the  sum  of  the  squares  on  the 
diagonals  and  four  times  the  square  on  the  join  of  the  mid- 
points of  the  diagonals. 

Data:  ABCD  the  4-side,  ^i^ the  join  of  the  mid-point 
of  the  diagonals  (Fig.  130). 

C 


Proof. 


whence 


Fig.  130. 

AR  ^Alr  =2AJ^'^  -^2 

^C'  +  C^'=2^^'+2C^'  (why?); 

AB^  4-  BC'  +  ~CIX  +  Djt 
=  ^BF^-{-2AF''  +  2  W 
=  ^BF^  +  /[AE-  +  ^EF'  (why?) 
^mf^-AC^'-^AEF"  (why?).  q.e.d. 


164  GEOMETRY.  [Th.  XCIX. 

Corollary.  The  sum  of  the  squares  on  the  sides  of  a  O 
equals  the  sum  of  the  squares  on  the  diagonals.     (Why?) 

1 88.  Theorem  XCIX.  — The  diffej-ence  between  the  squares 
on  a  side  of  a  symtnetric  A  and  on  the  join  of  the  vertex  to 
any  point  of  the  base  equals  the  rectangle  of  the  segments  into 
which  that  point  divides  the  base. 

Data  :  ABC  the  symmetric  A,  F  any  point  of  the  base 
(Fig.  130- 


P  A  .  P' 

Fig.  131. 

Proof.     Let  J/ be  the  mid-point  of  the  base,  then 

BT  =  BA'  4-  AB'  +2AP'  AM  (why  ?) . 

.-.  bP  -BA''  =  AF\AF+2AM\  =  AF'  PC.       q.e.d. 

Now  let  F  move  towards  ^4 ;  as  it  reaches  A,  the  tract 
AF  vanishes,  and  so  do  both  sides  of  the  equation.  As  F 
moves  on  to  F^  towards  C,  the  tract  AF  changes  sense,  it  is 
no  longer  reckoned  leftward,  but  rightward  ;  at  the  same 
time  the  left-hand  side  of  the  equation  chaiiges  sign,  BF 
becoming  less  than  BA ;  but  the  equation  still  holds,  for 
the  sign  of  AF  must  also  change  with  the  change  of  sense. 
We  are  yet  at  liberty  to  choose  the  difference  of  squares  as 
either  ^F" -K4^  or  ^A^ -'BF\  The  first  is  perhaps 
preferable,  and  we  see  that  when  F  is  without  the  tract  A  C, 
then  FA  and  FC  have  the  same  sense  and  sign,  being 
reckoned  the  same  way,  and  the  difference  is  positive ;  but 


Th.  C]  squares.  165 

when  P  is  within  A  C,  then  PA  and  PB  have  opposite  sense 
and  sign,  being  reckoned  oppositely ;  hence  we  may  say 
their  rectangle  is  negative,  and  accordingly  BP  is  less  than 
BA.  It  is  extremely  important  to  note  that  an  area  is  a 
sign- magnitude^  positive  or  negative  ;  it  has  sense. 

189.  When  P  is  at  A,  the  difference  BP'  —  Bj^  is  o  ;  as 
P  moves  towards  C,  the  tract  BA  remains  unchanged,  but 
BP  shortens  until  P  reaches  M \  thence  BP  lengthens  until 
it  again  becomes  equal  to  BA  or  BC,  as  P  falls  on  C. 
Hence  the  difference  BA  —  BP',  or  its  equal  PA  •  PC 
increases  while  P  moves  from  A  \.o  M  and  decreases  as  P 
moves  from  M  to  C  \  hence  it  is  greatest  when  P  is  at  M. 
A  value  of  a  variable  magnitude  that  is  thus  the  greatest 
within  a  series  of  successive  values,  or  that  is  greater  than 
the  values  just  before  it  and  just  after  it,  is  called  a  maxi- 
mum ;  while  a  value  that  is  less  than  the  values  next  before 
and  next  after  it,  is  called  a  minimum.  Hence  the  rectangle 
AP'  PC  is  a  maximum  for  Psit  M;  or  the  rectangle  on  the 
two  parts  into  which  a  given  tract  may  be  divided  is  a  maxi- 
mum when  the  parts  are  equal,  or  0/  all  rectangles  with  a 
given  perimeter,  the  square  is  the  maximinn.     Once  more, 


AC  =AP-\-  PC  =  AP-  +  PC  +  2  AP-PC. 
Now  ^  C  is  constant  while  P  moves  from  A  to  C,  and 
AP-  PC  is  greatest  when  P  is  at  M ;  hence  AP'  +  PC^  is 
least  when  P  is  at  M )  that  is,  the  sum  of  the  squares  on  the 
two  parts  of  a  given  tract  is  a  ?ninimum  when  the  parts  are 
equal 

190.  Theorem  C.  —  The  rectangle  of  the  distances  on  a 
secant  from  a  fixed  point  to  a  fixed  circle  is  constant  for  all 
secants. 

Data:    P\\\q.  fixed  point,  6*  the  fixed  circle  (Fig.  132). 


166  GEOMETRY.  [Th.  C. 

Proof.    Through  /*  draw  any  two  secants  cutting  the  circle 
at  A  and  B  and  at   C  and  D.     Then  in  the   symmetric 
AAOB,  OP--  '0A'=PA  '  PB  and  in  A  COD, 
~0P'  -~0C-  =  PC'PD  (why ?) . 

Hence  PA  •  PB  =  PC  -  PD  (why?),  no  matter  how  the 
secant  be  drawn  through  P.     q.  e.  d. 


Fig.  132. 

Def.  This  constant,  namely,  the  area  of  the  rectangle  of 
the  distances  from  the  fixed  point  to  the  fixed  circle  along 
any  secant,  is  called  the  power  of  the  point  as  to  the  circle. 

Corollary  i.  For  a  point  within  the  circle  the  power  is 
the  square  on  half  the  shortest  chord  through  the  point,  or 
on  half  the  chord  through  the  point  normal  to  the  radius 
through  the  point  (why  ?)  ;  for  a  point  without  the  circle 
the  power  is  the  square  on  the  tangent-length  from  the 
circle  to  the  point  (why?)  ;  for  a  point  on  the  circle  the 
power  is  zero  (why  ?) . 

Corollary  2.  The  power  of  a  point  without  the  circle  is 
positive  ;  of  a  point  within,  it  is  negative  (why?). 

Corollary  3.  If  PT'  —  power  of  P  as  to  S,  and  T  be 
on  S,  then  PT\^  tangent  to  S  (why?). 


Th.  CL] 


POWER-AXIS. 


167 


Corollary  4.  If  in  be  the  minimum  distance  from  the 
point  F  to  the  circle  6*  of  diameter  d,  then  the  power  of  the 
point  is  m(d  ±  m)  according  as  F  is  without  or  within  S. 
This  notion  of  the  power  of  a  point  as  to  a  circle  is  so  ex- 
ceedingly important  that  it  may  be  well  to  exemplify  its  use, 
in  passing,  though  not  necessary  for  our  present  purposes. 

191.  Theorem  CI. — All  points  having  equal  powers  as 
to  two  circles  lie  on  a  ray. 

Data:    6"  and  S  the  two  circles  (Fig.  133). 


Fig.  133. 


Proof.  Draw  a  normal  to  the  centre-tract  00^  (atiV), 
cutting  it  into  two  parts  d  and  d\  and  from  any  point  F  on 
this  normal  draw  tangents  FT,  FTK  Then  r  and  /  being 
the  radii, 

FN'--^d^=FT^+r',  and  FN'^-^t  d'''=F¥'^+ r^""  (why?). 

Hence         d'-  -  d'-  =  FT""-  FT''^-^  r^  -  r'\ 

Hence  FT  =  FT'^  when  and  only  wheii  d^—d^^  =  r^—r'^. 

If  then  we  find  iV^on  00',  dividing  it  so  that  d"^  —  d'-  = 
^2  _  ^'2^  ^jj^  ^i^jg  ^,^j^  always  be  done,  then  the  powers  of  all 


168  GEOMETRY.  [Th.  CI. 

points  on  the  normal  through  N  will  be   equal,   and  the 
powers  of  all  points  not  on   this  normal  will   be  unequal. 

Q.  E.  D. 

192.  Def.  This  most  important  ray  was  discovered  in 
18 13  by  Gaultier  and  named  by  him  i-adical  axis  of  the  two 
circles  ;  a  better  name  would  seem  to  be  power-axis.  This 
discovery  marked  and  in  a  measure  determined  the  renas- 
cence of  Geometry. 

Corollary  i.  The  common  secant  of  two  circles  is  their 
power-axis. 

Corollary  2.  The  common  tangent  of  two  circles  is  their 
power-axis. 

Corollary  3.  The  power-axes  of  three  circles  taken  in 
pairs  concur. 

Def.  The  point  of  concurrence  is  named  radical  centre 
or  power-centre  of  the  three  circles. 

Corollary  4.  A  circle  about  the  power-centre  with  the 
common  tangent-length  as  radius  intersects  the  three  circles 
orthogonally. 

The  importance  of  the  following  discussions  can  scarcely 
be  overestimated.  They  are  meant  to  ground  firmly  and 
in  strict  geometric  fashion  the  doctrine  of 

PROPORTION. 

193.  If  P  be  any  point  not  on  a  circle  S,  FT  a 
tangent,  and  FAB  a  secant  of  S,  then  we  have  seen  that 
FA'  FB  =  FT^  for  all  directions  of  FAB.  If  a  circle  / 
be  drawn  about  F  with  radius  FT,  it  will  cut  ^  orthogonally 
(why  ?)  ;  then  A  and  B  are  called  inverse  points  as  to  F, 
which  is  called  the  centre  of  inversion,  while  /  is  called  the 
circle  of  inversion. 


Th.  CIL]  proportion.  169 

194.  Theorem  CII  (converse  of  CI).  —  If  the  rectangle 
of  distances  from  a  point  to  two  points  on  a  ray  equal  {in  sense 
as  well  as  sign)  the  rectangle  of  the  distances  from  the  point  to 
tivo  poifits  on  another  ray,  both  rays  going  through  the  point, 
then  the  tivo  pairs  of  points  are  encyclic. 

Data :  P  the  point,  A  and  B,  C  and  D,  the  two  pairs  of 
points,  and  PA  •  PB  =  PC  -  PD  (Fig.  134). 


Proof.  The  circle  through  A,  B,  and  C  will  meet  PD 
somewhere,  as  at  D\  Then  PA  -  PB  =  PC  -  PD'  (why  ?)  ; 
hence  PD  =  PD'  (why?),  or  D'  is  D. 

Query.  Where  and  why  is  it  necessary  to  regard  the  sense 
of  the  rectangles  in  this  proof  ? 

195.  Now  let  us  consider  this  encyclic  quadrangle.  We 
know  that  the  opposite  inner  ^s,  as  A  and  D,  are  supple- 
mental (Fig.  135).  Think  of  the  plane  as  a  doubly  laid 
film  with  AC  drawn  in  the  lower  and  BD  drawn  in  the 
upper  layer,  and  imagine  PBD  taken  up,  turned  over,  and 


170 


GEOMETRY. 


[th.  cm. 


replaced  so  that  B  will  fall  on  B^  and  D  on  D' .  Then  will 
B^D^  be  II  to  AC.  For  the  inner  angle  at  Z>'  equals  inner 
angle  at  D ;  hence  the  inner  ^s  at  A  and  Z>'  are  supple- 
mental (why?)  ;  hence  B^D^  and  AC  sue  II. 


Fig.  135. 

This  operation  of  turning  over  BD  into  the  position  B'D' 
we  may  call  inverting  BD. 
Hence 

Theorem  CIII.  — If  one  side  of  an  encyclic  quadrangle  be 
inverted^  the  resulting  figure  is  a  trapezoid. 
Conversely, 

196.  Theorem  CIV.  — If  07ie  parallel  side  of  a  trapezoid 
be  inverted  J  the  figure  resulting  is  an  encyclic  quadrangle. 
The  ready  proof  is  left  for  the  student. 

We  note  that  the  proofs  hold  as  well  for  the  crossed  quad- 
rangle and  trapezoid  as  for  the  convex  or  normal. 

197.  Now  let  PAC  and  PB^D^  be  any  two  A  with  com- 
mon vertical  angle  P  and  II  bases  A  C  and  B^D  ;  then 

PA  .  PB'  =  PC  .  PD'  (Fig.  135). 


Th.  civ.]  proportion.  171 

Proof.  For  on  inverting  B^ D^  into  BD  the  quadrangle 
ABCD  is  encyclic.  Hence  PA  •  FB  =  PC  -  PD,  and  PB 
=  PB',  PD  =  PD\     Hence  q.  e.  d. 

198.  Conversely,  Let  PAC  and  PB^D^  be  any  two  A 
with  common  vertical  ^  and  let 

PA  '  PB' =  PC  •  PD\ 

Then  AC  and  B'D'  are  II. 

Proof.  Draw  through  A  a  II  ^^C  to  B'D\  cutting  PB'  at 
C;  then 

PA  .  PB'  =  /^r  •  PD\ 

Hence  /'C:= /'C  (why?).  q.e.d. 

199.  These  relations  are  very  simple  and  easy  of  com- 
prehension, but  their  statement  in  words  is  very  awkward  and 
cumbrous.  To  relieve  the  difficulty  of  verbal  expression  we 
introduce  a  new  arbitrary  definition  and  a  new  arbitrary 
symbolism. 

Def.  If  the  rectangle  of  two  tracts  equals  the  rectangle 
of  two  other  tracts,  the  four  tracts  are  said  to  be  in  propor- 
tion, or  to  form  a  proportion,  or  to  be  proportional. 

Symbolism.  If ;/  and  v  be  the  one  pair,  x  and  J^'  the  other 
pair  of  tracts,  then  we  write  u  :  x:  :y  :  v  and  read  u  is  to 
X  as  y  is  to  v. 

200.  In  order  to  speak  readily  about  this  proportion  we 
define  further : 

Definition  i.  The  tracts  are  called  terms  of  the  propor- 
tion. 

Definition  2.  The  first  and  last  are  called  extremes;  the 
second  and  third  are  called  means. 


172  ^        GEOMETRY.  [Th.  CIV. 

Definition  3.  The  first  and  second  are  called  the  first 
couplet ;  the  third  and  fourth,  the  second  couplet. 

Defifiition  4.  The  first  and  third  terms  are  called  antece- 
dents; the  second  and  fourth  are  called  consequents. 

Definition  5.  When  the  two  means  are  equal,  each  is  called 
the  mean  proportional  or  geometric  mean  of  the  extremes. 

Definition  6.  The  fourth  term  is  called  the  fourth  propor- 
tional to  the  other  three  taken  in  order ;  or,  if  the  means  be 
equal,  it  is  called  a  third  proportional  to  the  other  two  taken 
in  order. 

Definition  7.  When  the  means  are  exchanged,  or  the 
extremes  are  exchanged,  the  proportion  is  said  to  be  alter* 
nated. 

Definition  8.  When  the  terms  of  each  couplet  are  ex' 
changed,  the  proportion  is  said  to  be  inverted. 

Definition  9.  When  in  place  of  the  first  or  second  term 
of  each  couplet  is  put  the  sum  (or  difference)  of  the  terms 
of  that  couplet,  the  proportion  is  said  to  be  compounded  (or 
divided) . 

201.  Since  by  a  proportion  we  mean  nothing  more  and 
nothing  less  than  that  the  rectangle  of  the  means  equals  the 
rectangle  of  the  extremes,  it  is  plain  that  the  same  proportion 
may  be  written  in  several  different  ways,  thus  : 

u  \  X  \  \v  '.  y,  u  \  v\ '.  X  :  y,  y\  X  \  w  \  u,  y  \v  \  \  X  \  u, 
X  \u  \\y  w,  X'.ywu'.v,  v  \  u  :  \y  \  x,  v:  y  \  \  u  :  x, 

all  mean  precisely  the  same ;  namely,  rectangle  of  //  and  y 
=  rectangle  of  v  and  x. 

202.  All  of  these  forms,  and  no  others,  may  be  derived 
from  any  one  of  them  by  alternation  and  inversion  ;  hence 


Th.  CVL] 


PROPORTION. 


173 


Theorem  CV.  —  When  four  tracts  are  in  proportion  they 
are  in  proportion  by  alternation  and  by  inversion  {alternando 
et  invej'tendo). 

The  simpUfication  of  expression  will  now  soon  become 
evident.     We  must  still  further  premise,  however, 

Definition  lo.  When  the  angles  of  one  A  are  respectively 
equal  to  those  of  another,  the  A  are  said  to  be  mutually- 
equiangular,  and  the  sides  opposite  equal  angles  are  said  to 
correspond,  as  do  also  the  equal  angles  themselves. 

203.  Theorem  CVI.  —  Corresponding  sides  in  two  mutu- 
ally equiangular  A  aj'e  proportional  in  pairs. 

Data  :    ABC,  A^B'C  the  A,  A  =  A',  B  =  B',  C=  C\ 

Proof.    Fit  ^  ^  on  ^  ^' ;  then  ^C  is  II  to  B' O  (why?). 

Hence  AB  'A'C'  =  A'B'  •  AC  (why?), 

or  AB:  A'B'::  AC:  A' a. 

C 


Fig.  136. 

Similarly,  by  putting  B  on  B',  C  on  C, 
BC:B'C  ::BA:B'A', 
CA  :  a  A'  ::CB:  CB', 


Q.  E.  D. 


204.    These  three  proportions  may  be  conveniently  written 
as  a  continued  proportion,  thus  : 


174 


GEOMETRY. 


[Th.  evil. 


AB'.BC'.CA::  A'B'  :B'C':  C'A',  read 

AB  is  to  BC  is  to  CA  as  A'B'  is  to  B'C  is  to  C'A' ; 

or  perhaps  still  better  thus  : 

AB  :  A'B'  ::BC:B'C'::CA:  C'A',  read 

AB  is  to  A'B'  as  BC  is  to  B'C  as  C^  is  to  C'A' ; 

they  are  exactly  equivalent  to  the  three  equations 

AB.B'C'  =  A'B'  'BC,     BC-  C'A' =  B'C-  CA, 
CA-A'B'=  C'A'  -AB. 

205.   Theorem  CVII.  —  Conversely,  Two  Awt'fk  sides  pro- 
portional are  fnutually  equiangular. 

Data:    ^^C,  ^'.5' C  the  two  A,  and 

AB.A'B'.-.BC'.B'C'.'.  CA-.  C'A'  (Fig.  137). 
C  C 


Proof.     From   C  lay  off  CA  equal    to   CA'  and    draw 
A^B^  II  to  AB.     Then 

CA^ '  CB=CA'  CB^  (why?). 
But  CA' '  CB=CA'  CB'  (why?). 

Hence  CA  -  CB^  =  CA  -  CB'  (why?). 

Hence  CB,  =  CB'. 

Similarly,  A^B,  =  A'B'. 


Th.  CIX.] 


PROPORTION. 


175 


Hence  A^B^C  and  y^j^iCare  congruent  (why?). 
Hence  A^B^  C  and  AB C  are  mutually  equiangular  (why  ?) . 

Q.  E.  D. 

Thus  it  appears  that  mutual  equiangularity  and  propor- 
tionality of  sides  coexist  and  imply  each  other.  Two  such 
A  mutually  equal  in  their  angles  and  proportional  in  their 
sides  are  called  similar. 

206.  Theorem  CVIII.  —  T^vo  A  having  an^of  one  equal 
to  an  ^  of  the  other  and  the  including  sides  proportional  are 
similar. 

Data  :    ABC,  A'B'C  the  two  A, 

^C=  ^a,  and   CA  :  C'A' :  :  CB  :  C'B'  (Fig.  138). 


Proof.  Fit  C  on  C;  then  by  the  proportion  A'B'  is  II  to 
AB.     Hence,  etc.     q.  e.  d. 

207.  Theorem  CIX.  —  Tiifo  A  having  two  pairs  of  sides 
proportional,  and  a  pair  of  angles  opposite  the  larger  sides  in 
each  equal,  are  similar. 

Data :    ABC,  AB^C  the  two  A, 

AB>BC,A'B^>B'a. 

AB\A'B'  wBC'.B'C,  and  ^C=^C'  (Fig.  139). 


176 


GEOMETRY. 


[Th.  ex. 


Proof.     Fit  ^  C  on  C  ;  then  since 

A'B'  >  B^C  and  AB  >  BC,  both  ^^  and  AB' 

must  be  drawn  making  the  inner  ^s  at  A  and  A^  acute. 
From  the  proportion,  AB  and  ^'Z?'  are  now  li  (prove  it)  ; 
hence  the  A  are  mutually  equiangular  and  hence  similar. 

Q.  E.  D. 


N.B.  If  AB  were  <  BC  and  A'B'  <  B'C,  then  AB  and 
A'B'  might  make  the  ^s  at  A  and  ^'  supplemental  instead 
of  equal,  and  hence  AB  and  A^B^  anti-ll  instead  of  II,  in 
which  case  the  A  would  not  be  similar.  Dra^  a  figure 
illustrating  this  case. 

Compare  the  conditions  of  similarity  with  the  conditions 
of  congruence  between  two  A. 

208.  Theorem  CX.  — If  two  proportions  agree  in  the  first 
three  terms  of  each,  they  agree  ifi  the  fourth  also. 

Data  :     u  :x:  -.7! :  y,  and  u\  x\\v\  y'. 

Proof,     uy  =  vx,  and  uy'  =  vx  :  hence  uy  =  uy\ 

Hence   j'  =  jv'  (why?),     q. e. d. 

209.  Theorem  CXI. — If  two  proportions  agree  in  one 
couplet,  the  other  couplets  form  a  proportion. 


Th.  CXIL]  proportion.  177 

Data :    u\x'.'.v\y\  u-.xww.z  (Fig.  140) . 

Proof.  Suppose  w  <  v  and  on  two  half-rays  through  P 
lay  oKFU,  FX,  FV,  and  FY:=  u,  x,  v,y.  Open  the  angle 
at  F  until  UV=  w.  This  is  possible,  since  w  <v.  Draw 
XY and  call  it  s'.  Then  since  w. x:\v\y,  w  and  s'  are  il ; 
hence  u\x\'.w\z\  Hence  s'  =  s;  (why  ?) ,  and  hence  v.yw 
w  :z^  ox  z  (why  ?) .     q.  e.  d. 

V 


Fig.  140. 

Corollary.  We  may  write  the  three  proportions  as  a  con- 
tinued proportion,  thus : 

u:x:\v:y::7V'.z. 

210.  Theorem  CXII. — If  four  tracts  are  in  proportion, 
they  are  in  proportion  by  composition,  and  by  division,  and 
by  composition  and  division  (  Componendo,  dividendo,  corn- 
pone  ndo  et  dividendo) . 

Datum  :  u  :  x  -.-.v-.y. 

Proof.  On  any  pair  of  half-rays  through  any  point  F  lay 
o^  FU=  u,  FV=  V  ;  from  ^lay  off  on  /*  a  tract  UX=  x, 
and  on  a  li  to  FV  a  tract  UY=y.     Then  the  A  FUV  and 


178 


GEOMETRY. 


[Th.  CXIIL 


UXY  are  similar  (why?)  ;  hence  UV  and  XY  are  II 
(why?),  and  VR  is;'  (why?).  Also  A  PUV  and  PXR  are 
similar  (why?).  Hence  u -.u -\-  xwv  \v  -\- y  {^  Componendo). 
Again,  from  P  lay  off  as  before  PU=^  u,  PV=  v,  PX=^  x, 
PV=y;  then  Jry and  ^Fare  II  (why?)  (Fig.  141). 


Fig.  141. 

Draw  XR  II  to  PV;  then  A  PUV  and  XUR  are  similar 
(why?).     And  XR  =  v-y  (why?),  XU=  u-x. 

Hence  u\u—x\\v\v—y  {Dividendo). 
Hence  ti -\- x\  u  —  x -.  w -\- y  :v —y    (why?)    {Componendo 
et  dividendo). 

*2ii.  Theorem  CXIII. —  Whe7i  four  trac is  are  in  propor- 
tion, equimultiples  of  either  or  both  couplets  are  in  proportion. 

Datum  :  u  :  x  \  \  v.  y. 

Proof.  By  Def  rect.  uy  =  rect.  vx.  Hence  m  (rect.  uy) 
=  m  (rect.  vx). 

But  m  (rect.  uy)  —  rect.  mt/-y  and  m  (rect.  vx)  =  rect. 
v-mx ;  hence  mu-y  =  v-mx,  or  mu  :  mx  :  :  v  :y. 

Similarly,  nv  :  ny:  :  v  :  y. 

Hence  mu  :  mx  : ;  nv  :  ny.  q.  e.  d. 


Th.  CXIV.]  proportion.  179 

Corollary.  The  multipliers  ;;/  and  n  may  be  disposed  any 
way  in  the  proportion  provided  only  that  each  appears  in  a 
mean  and  in  an  extreme. 

212.  Def.  Two  tracts  are  said  to  be  divided  similarly 
when  all  the  parts  of  the  one  and  all  the  parts  of  the  other 
taken  in  the  same  order  form  a  continued  proportion ;  thus, 
if  a,  bj  c,  d  be  the  parts  of  one  and  a\  b\  c\  d^  the  parts  of 
the  other,  and  a  :  b  :  c  \  d  w  a'  -.  b^ :  c'  \  d\  then  the  divison  is 
similar.  Two  parts  forming  a  couplet,  as^  and  a\  are  said 
to  correspond. 

213.  Theorem  CXIV.  —  Two  transversals  of  a  system  of 
parallels  are  divided  similarly  by  the  parallels. 

Data :  AA\  BB\  etc.,  the  lis,  PT  and  PT^  the  trans- 
versals (Fig.  142). 

P 


D' 


Fig.  142. 
Proof.    From  similar  A, 

PA  -.PA'i'.PB:  PB'  ..PC'.PC'.'.PD'.  PD\ 
Hence,  alternando  and  dividendo, 
PA  \PA'  wAB'.  A'B' :  :  BC.B'C  ..CD:  CD\  q.e.d. 


180 


GEOMETRY. 


[Th.  CXV. 


Corollary  i.   The  intercepts  of  the  lis  are  proportional  to 
their  distances  from  the  vertex  P. 

Corollary  2.   A  system  of  ||s  divides  the  rays  of  a  pencil 
similarly  (Fig.  143). 


Fig.  143. 

Corollary  3.  The  intercepts  of  the  rays  on  any  two  lis  are 
proportional. 

*2i4.  Theorem  CXV  (Ptolemy's).  —  In  an  encyclic  quad- 
rangle the  rectangle  of  the  diagonals  equals  the  sum  of  the 
rectangles  of  the  opposite  sides. 

Datum  :    ABCD  an  encyclic  quadrangle. 


Fig.  144. 


th.  cxvi.]  proportion.  181 

Proof.  Draw  the  diagonals  and  also  AN,  making  ^  AND 
=  ^ABC.  Then  A  ADN zx\^  ACB  are  similar  (why?) ; 
hence  BC-AD^AC-DN  (Fig.  144). 

Also         BNA  and  CD  A  are  similar  (why?)  ; 
hence  AB'CD  =  AC'BN 

On  addition  there  results 

AB'CD-{-BC  •DA  =  AC- BD.  q.e.d. 

*2i5.  Theorem  CXVI.  —  The  j^ec tangle  of  two  sides  of  a 
A  equals  the  rectangle  of  the  altitude  to  the  third  side  and 
the  circum-diameter. 

Data:    ABC  the  A,  6* the  circumcircle  (Fig.  145). 


Proof.    A  ABD  and  £BC  are  similar  (why?). 
Hence  AB  :  EB  :  :  BD  :  BC, 

or  AB'BC=EB  'BD.  q.e.d. 

216.    Def.  When  a  tract  is  divided  into  two  parts  propor- 
tional to  two  other  tracts,  it  is  said  to  be  divided  in  the  ratio 


182  GEOMETRY.  [Th.  CXVII. 

of  those  tracts,  or  the  ratio  of  the  parts  is  said  to  equal  the 
ratio  of  the  tracts. 

N.B.  This  is  a  definition  of  equality  of  ratios,  but  not  of 
ratio  itself;  this  latter  we  now  neither  need  nor  attempt  to 
define,  but  we  write  it  thus,  /  :  m,  and  read  ratio  of  I  to  m. 

Def.  The  division  may  be  inner,  when  the  dividing  point 
P  falls  within  the  tract,  or  outer ^  when  it  falls  without  the 
tract. 

217.  Theorem  CXVII. — A  tract  fnay  be  divided  innerly  and 
outerly  in  any  given  ratio,  but  in  each  case  at  only  one  point. 

Data:  AB  the  tract  to  be  divided,  /  and  in  the  other 
tracts. 

Proof.  I.  From  A  draw  any  half- ray  ;  lay  off  on  it  AL  —I 
and  LM  =  m  ;  join  BM,  and  draw  PL  II  to  it  (Fig.  146). 


Fig.  146. 

Then  AP  :  PB  :\  I :  m  (why?)  ;  hence  P  divides  AB 
innerly  in  the  given  ratio. 

Again  supposing  in  <  I,  lay  off  LM  backwards  towards  A  ; 
join  BM  and  draw  LR  II  to  it. 

Then  AQ:  QB  \  :l :  m  (why?)  ;  hence  Q  divides  AB 
outerly  in  the  given  ratio. 


Th.  CXVIIL]  proportion.  183 

Proof.  2.  If  jP'  be  any  point  of  division  of  AB  in  the 
ratio  /:  m,  then  Z/"  is  II  to  MB  (why?)  ;  but  there  is  only 
one  II  to  AB  through  L ;  hence  /"  falls  on  F,  i.e.  there  is 
only  one  point  of  inner  division  in  the  ratio  /:  m.  Similarly, 
show  that  there  is  only  one  point  Q  of  outer  division  in  the 
ratio  /:  7n.     q.e.d. 

218.  N.B.  I.  In  case  of  inner  division  the  parts  AP, 
PB  are  reckoned  the  same  way,  both  rightward ;  in  case  of 
outer  division  the  parts  A  Q,  QB  are  reckoned  oppositely, 
one  rightward,  the  other  leftward. 

2.  In  speaking  of  the  ratio  of  the  tracts  /  and  m  the 
order  of  mention  is  essential ;  the  ratio  of  /  and  m  (what- 
ever it  may  be)  is  not  the  same  as  the  ratio  of ;//  and  /.  So, 
too,  the  order  of  mention  of  the  ends  of  the  tract  AB  is 
essential :  we  mean  that  the  first  part  is  to  be  reckoned /r^»« 
A  and  the  second  part  fo  B ;  to  divide  AB  in  a  given  ratio 
is  not  the  same  as  to  divide  BA  in  that  ratio. 

Def.  When  a  tract  AB  is  divided  innerly  and  outerly  at 
P  and  Q  in  the  same  ratio,  it  is  said  to  be  divided  harmoni- 
cally, A  and  B  are  said  to  be  harmonically  conjugate  with 
P  and  Qf  A  and  B,  P  and  Q  are  said  to  form  two  harmonic 
pairs,  and  the  four  points  A,P,B,  Q,  taken  in  order,  are  said 
to  be  four  harmonic  points  or  to  form  an  harmonic  range. 

219.  Theorem  CyiYlll.—If  P  and  Q  divide  AB  har- 
monically, then  A  and  B  divide  PQ  harmonically. 

Data  :  AB  a  tract,  P  and  Q  the  points  of  inner  and  outer 
division  in  any  ratio,  as  /:  w. 


Proof.  AP'.PB 
hence  AP-.  BP 
or  PAiAQ 


I :  m,  and  A  Q  :  QB  \\  l\m\ 
AQ:  QB  (why?), 
PB  :BQ  (why?).  q.e.d. 


184  GEOMETRY.  [Th.  CXIX. 

220.  N.B.  I.  To  the  inner  and  outer  division  oi  AB  by 
P  and  Q  corresponds  the  outer  and  inner  division  of  PQ 
by  A  and  B. 

2.  The  term  harmonic  is  borrowed  from  the  theory  of 
musical  intervals ;  four  tracts  a,  b,  c,  d  are  said  to  be  har- 
monically or  musically  proportional  when  the  first  is  to  the 
last  as  the  difference  of  the  first  two  is  to  the  difference  of 
the  last  two  ;  i.e.  when   a-.  d\\  a  —  b  :  c  —  d. 

Now  let  the  student  prove  that  '\i  AP\  PBw  AQ\  QB, 

then  AP:  QB::  AP-  PB:AQ-QB, 

and  so  justify  the  use  of  the  term  harmonic. 

221.  Theorem  CXIX.  —  The  inner  and  outer  mid-rays  of 
an  angle  of  a  IS  divide  the  opposite  side  harmonically  in 
the  ratio  of  the  adjacent  sides. 

Data :  CP  and  CQ,  the  inner  and  outer  mid-rays  of  the 
angle  C  of  the  A  ABC,  meeting  the  side  AB  at  P  and  Q 
(Fig.  147). 


Fig.  147. 

Proof.  Draw  BD  II  to  CP;  then  <C  DBC=^  BCP 
(why?)=  ^  PC  A  (why?)=  ^  CDB  (why?)  ;  hence  CD 
=  CB. 


Th.  CXX.] 


PROPORTION. 


185 


Also  AF'.PB:.AC 
Similarly,  AQ:QB 
Hence  AP:  PB 


CD  (why?)  or  AP:PB::AC:  CB. 

'.AC'.CB. 

:AQ:  QB::AC:  CB.  Q.E.D. 


Corollary.  Conversely,  if  two  rays  divide  a  side  of  a  A 
innerly  and  outerly  in  the  ratio  of  the  adjacent  sides,  they 
are  the  inner  and  outer  mid-rays  of  the  opposite  angle  (why?). 

222.  Theorem  CXX. — If  a  normal  be  drawn  from  the 
vertex  of  the  right  angle  in  a  right  A  to  the  hypotenuse^  then 

I.  The  A  will  be  cut  into  two  right  A  similar  to  each 
other  and  to  the  original  A. 

II.  The  ?iormal  tract  will  be  a  mean  proportional  between 
the  segments  of  the  hypotenuse. 

III.  Each  side  of  the  right  angle  will  be  a  mean  pro- 
portional between  the  whole  hypotenuse  and  the  adjacent 
segment. 

Data:  ABC  the  right  A,  C^  the  normal  to  the  hypote- 
nuse (Fig.  148). 

C 
1" 


Fig.  148. 

Proof.    I.   The  A  ABC,  ACJV,  and  BCN  are  plainly 
mutually  equiangular  (why?)  and  hence  similar,     q.e.d. 

Q.E.D. 


Q.  E.  D. 


II.   Hence  AN 

CN: 

:  CN 

NB. 

III.   Also     AB: 

AC: 

:  AC 

AN, 

and                   AB 

BC: 

'.BC . 

BN. 

186 


GEOMETRY. 


[Th.  CXXI. 


Corollary.  Conversely,  if  a  tract  CiVfrom  the  right  angle 
divides  the  A  into  similar  A,  or  is  a  mean  proportional 
between  the  segments  of  the  hypotenuse,  or  divides  the 
hypotenuse  so  that  either  side  is  a  mean  proportional 
between  the  whole  hypotenuse  and  the  adjacent  segment, 
then  it  is  normal  to  the  hypotenuse. 

223.  Theorem  CXXI.  — If  four  concurrent  rays  {or  rays 
of  a  pencil^  cut  one  transversal  harmonically,  they  cut  every 
transversal  harmonically. 

Data :  Any  transversal  cut  harmonically  at  A,  B,  C,  D 
by  four  rays  concurrent  in  O  and  PQRS  any  other  trans- 
versal (Fig.  149). 


Fig.  149. 

Proof.    Draw  through  two  conjugate  points  as  B  and  D 
any  two  transversals  II  to  PQRS ; .  then 


AB- 

BC: 

:    AD 

DC, 

or 

AB'. 

AD: 

:    BC 

DC  (why?). 

Also 

AB'. 

AD: 

AB 

A^D, 

and 

BC: 

DC: 

BC 

DC  (why?) 

Th.  CXXIL] 

HARMONIC  POINTS. 

hence 

A^B 

A^^D.'.   BC:DC\ 

or 

A'B 

BC--  A^D-DC\ 

But 

A'D 

DC"::  A^D':D'C'  (why?) 

hence 

A^B 

BC::  A^D':D^C; 

hence 

PQ 

:    QR::    PS  :  SR  (why?). 

187 


Q.  E.  D. 


Corollary.  The  proportion  AB  :  BC:  :  AD  :  DC  is  not 
affected  by  any  movement  of  O,  while  A,  B,  C,  D  remain 
fixed  ;  neither,  then,  is  the  proportion  PQ  :  QR  :  :  PS :  SQ. 

224.  Theorem  CXXII.  —  A  chord  of  a  circle  and  the 
tangents  at  its  ends  cut  the  conjugate  diameter  harmonically 
(Fig.  150J. 

Data :  S  the  circle,  TV  the  chord,  PT,  PT  tangents  at 
its  ends,  PA  the  conjugate  diameter. 


Fig.  150. 

Proof.    TB  bisects  ^  PTD  innerly  (why?)  ; 
hence         TA  bisects  it  outerly  (why  ?)  ; 
hence        A,  D,  B,  Pare  four  harmonic  points  (why?). 


Q.  E.  D. 


188  GEOMETRY.  [Th.  CXXIII. 

SIMILAR   FIGURES. 

225.  We  have  already  found  that  mutually  equiangular 
A  have  their  corresponding  sides  proportional,  and  con- 
versely, and  we  have  named  such  A  similar.  A  more  gen- 
eral notion  of  similarity  may  be  obtained  thus  : 

Let  two  points,  P  and  P,  move  at  will  in  the  plane,  but 
under  these  restrictions  : 

1 .  The  ray  PP  shall  pass  always  through  a  fixed  point  O. 

2.  The  proportion  shall  always  hold  OP:  OP^w  t\  /', 
where  /  and  /'  are  any  two  fixed  tracts ;  then  the  paths  of 
P  and  P^  are  called  similar  figures  similarly  placed  (or 
homothetic) . 

The  point  O  is  called  the  centre  of  similitude  ;  outer ^  if 
O  divides  PP  outerly  ;  inner,  if  innerly. 

226.  If  an  eye  were  placed  at  the  outer  centre,  it  would 
manifestly  see  the  one  figure  through  the  other,  point  for 
point ;  hence  the  two  figures  are  said  to  be  in  direct  per- 
spective ;  if  6>  be  the  inner  centre,  they  may  be  said  to  be  in 
indirect  perspective,  or  in  contra-perspective. 

If  on  any  ray  through  O  there  be  taken  two  points,  C  and 
C\  such  that  0C\  OC  \\t\t\  then  C  and  C  are  said  to 
correspond  to  each  other  with  respect  to  the  centre  of  simili- 
tude O  in  the  ratio  of  similitude  / :  /' ;  any  tract  between 
two  points  in  the  one  figure  is  said  to  correspond  to  the 
tract  between  the  corresponding  points  in  the  other  figure. 

227.  Theorem  CXXIII.  —  Coi-respondent  tracts  in  two 
perspective  figures  are  parallel  and  in  the  ratio  of  similitude 
to  each  other. 

Data :  O  the  centre,  A  and  A\  B  and  B^  two  pairs  of 
corresponding  points  (Fig.  151). 


Th.  cxxv.j 


SIMILAR   FIGURES. 


189 


Proof.    The  A  A  OB  and  A^OB'  are  similar   (why?); 
hence  AB  and  A'B^  are  II,  and  AB :  A^B'  ::t:t'   (why?). 

Q.  E.  D. 


P' 


Fig.  151. 

228.  Theorem  CXXIV.  —  Conversely,  If  from  two  points, 
A  and  A',  correspondent  as  to  O,  there  be  laid  off  two  II 
tracts  AB,  AB^  in  the  ratio  OA  :  0A\  then  B  and  B'  cor- 
respond.    Let  the  student  give  the  proof. 

229.  Theorem  CXXV. — Any  two  circles  are  in  perspec- 
tive and  contra-perspective. 

Data:    6"  and  S  any  two  circles  (Fig.  152). 

Proof.  Divide  the  centre  tract  CO  innerly  and  outerly  in 
the  ratio  of  the  radii  r :  r'  at  points  /  and  O.  Draw  any 
secant  OA,  and  on  it  lay  off  OA'  so  that 

OC:  OC::  OA  :0A\ 

The   A    OCA   and    OCA'  are   similar   (why?).      Hence 
C'A'  =  r'  (why?)  ;  hence  A'  is  on  S' ;  hence  any  point  of 


190 


GEOMETRY. 


[Th.  CXXVI. 


.S  has  its  correspondent  on  6"'  in  the  same  fixed  ratio  of  the 
radii ;  hence  6*  and  S  are  similar,  and  are  plainly  in  perspec- 
tive. For  /  the  reasoning  is  the  same,  but  the  tracts  being 
laid  off  oppositely,  the  figures  are  in  contra-perspective. 

Q.  E.D. 

Coi'ollary.    Common  tangents  to  the  two  circles  go  each 
through  a  centre  of  simiHtude. 


Fig.  152. 

230.  Theorem  CXXVI.  —  Conversely,  Atiy  figui-e  simi- 
lar to  a  cii'cle  is  itself  a  circle. 

Data  :  S  a  circle,  S  similar  to  it  with  respect  to  the  cen- 
tre of  similitude  O,  in  the  ratio  r :  r'. 

Proof.  Find  the  corresponding  point  C  of  the  centre  C 
of  S,  and  draw  through  O  any  secant  meeting  S  and  S  in 
the  corresponding  points  A  and  A\  Then  triangles  COA 
and  COA'  are  similar  (why?)  ;  hence 

OC:  OC::  CA  :  C'A', 


Th.  cxxviii.]        similar  figures.  191 

and  since  (9C,  OC ,  and  CA  are  constant  in  length,  so,  too, 
is  CA\  Hence  S  is  a  circle  about  C  as  centre.  The  like 
proof  holds  for  the  inner  centre  /.     q.  e.  d. 

231.  Theorem  CXXVII.  —  The  angle  between  two  tracts 
in  one  figure  equals  the  angle  between  the  corresponding  tracts 
in  any  similar  figure. 

Data:  i^and  F^  (Fig.  151),  two  similar  figures  similarly 
placed.  AB  and  BC,  two  tracts  in  F.  A'B'  and  B'C,  the 
corresponding  tracts  in  F'. 

Proof.  Draw  OA,  OB,  OA',  OB' ;  then  the  theorem  fol- 
lows at  once  from  similar  A.  But  if  the  figures  be  not 
similarly  placed,  and  F"  be  one  of  them  congruent  with  F', 
suppose  F"  brought  to  coincidence  with  F' ;  then  what  has 
just  been  proved  for  F'  holds  for  F".     q.e.d. 

Corollary.  F"  may  be  brought  to  coincide  with  F'  by 
being  mtxtXy  pushed,  —  all  rays  remaining  parallel  to  them- 
selves in  their  original  position,  until  one  point  of  7^"  falls 
on  the  corresponding  point  F\  —  and  then  being  merely 
turned  until  another  point  of  /^"  falls  on  its  correspondent  in 
F'.  If  the  figures  still  do  not  coincide  throughout,  but  only 
on  the  common  ray  through  three  points,  it  will  be  necessary 
and  sufficient  to  revolve  F'  about  that  common  ray  through 
a  straight  angle,  which  revolution  will  change  opposition  into 
superposition  of  the  figures.  In  this  revolution  all  rays  in 
the  figure  are  turned  through  the  same  angle ;  hence 

232.  Theorem  CXXVIII. — /;/  two  similar  figures  all 
lines  are  inclined  to  their  corresponde7its  under  the  same 
angle.  Perhaps  we  might  name  this  angle  the  anomaly  of 
the  one  figure  as  to  the  other. 

In  two  similar  figures  point  corresponds  to  point,  angle  to 


192  GEOMETRY.  [Th.  CXXIX. 

equal  angle,  tract  to  tr^ct,  in  the  same  ratio ;  hence  it  is 
plain  that 

233.  Theorem  CXXIX. — Any  two  similar  figures  7nay  be 
cut  up  into  pairs  of  similar  figures  in  the  sa?fie  ratio  of  simili- 
tude and  order  of  arrangement. 

INSTRUMENTS. 

234.  There  are  four  important  instruments  used  in  prac- 
tice in  the  construction  of  similar  figures :  proportional 
compasses,  sector,  diagonal  scale,  and  Pantagraph  or  Eido- 
graph.  Of  these  the  last  is  the  most  interesting  and  il- 
lustrates in  its  working  very  accurately  the  definition  given 
above  of  similar  figures  in  contra-perspective.  Every  well 
appointed  academy  should  be  furnished  with  these  instru- 
ments, which  may  easily  be  explained  and  operated. 

CONSTRUCTIONS. 

235.  The  doctrine  of  proportion  is  extensively  employed, 
not  only  in  mechanical  drawing  with  the  instruments  men- 
tioned, but  also  in  the  strict  logical  solution  of  problems  of 
construction. 

236.  Problem  I.  —  To  divide  a  tract  {innerly  and  outerly) 
in  a  give  ft  ratio,  as  of  I:  m.     (See  p.  182.) 

237.  Problem  II.  —  To  divide  a  tract  {innerly  and  outerly) 
into  any  number  of  parts  proportional  to  /,  ;//,  n,  p  '  •  •. 

Solution.  From  the  beginning  of  the  tract  AB  draw  any 
half-ray,  as  AR;  on  it  lay  off  in  order  consecutively  the 
tracts  /,  m,  n,  p,-  •  • .  Join  the  end  of  the  last  with  the  end  of 
AB,  and  through  the  ends  of  the  others  draw  parallels ;  to 
this  join  RB ;  they  divide  AB  as  required  (why?). 

Let  the  student  solve  the  problem  of  outer  division. 


CONS  TR  UC  TIONS. 


193 


238.  Problem  III. 
fwo  tracts. 


To  construct  the  geometric  fnean  of 


Solution.  On  the  sum  of  the  two  tracts,  /and  m  (Fig.  153), 
as  diameter,  draw  a  circle,  and  through  their  common  point 
draw  a  half-chord  conjugate  to  the  diameter ;  it  is  the  mean 
proportional  required  (why?). 


Fig.  153. 

239.  Problem  IV.  —  To  construct  a  square  equal  to  a 
given  rectangle.     Proceed  as  in  Problem  III. 

240.  Problem  V.  —  Knowing  one  dimension  of  a  rectangle 
equal  to  a  given  rectangle,  to  find  the  other;  or,  given  three 
tracts,  to  find  a  fourth  proportional  to  them  in  order. 


f  IG.  154. 

Solution.    Prolong  one  side  of  the  given  rectangle  by  the 
given  side  of  the  other,  as  AB  to  Qy  and  draw  QC  meeting 


194 


GEOMETRY. 


AD  at  R;  then  DR  is  the  other  dimension  sought  (why?) 
(Fig.  154).     Or, 

On  any  two  half-rays  meeting  at  A  lay  off  AB  and  AD 
equal  to  the  given  sides  or  the  second  and  third  of  the  three 
tracts,  and  on  either,  as  AD,  lay  off  AQ  equal  to  the  one 
given  dimension  or  the  first  tract.  Draw  BQ  and  DR  II  to 
BQ;  then  AR  is  the  fourth  proportional  sought  (why?). 

These  constructions  require  us  either  to  know  the  angle  at 
A  or  else  to  draw  a  parallel.  But  we  may  proceed  thus,  avoid- 
ing all  use  of  pai^allels  and  angles :  draw  a  large  circle  and 
lay  off  as  a  chord  of  it  the  difference  of  the  second  and  third 
proportionals  ;  from  the  ends  A  and  B  of  this  chord  lay  off 
AP  and  BP  equal  to  the  second  and  third  proportionals ; 
about  P  describe  a  circle  with  the  first  proportional  as  radius 
intersecting  the  circle  at  /;  draw  PI,  meeting  the  circle 
also  aty;  then  ^is  the  fourth  proportional  sought  (why?) 
(Fig.  155)- 


Fig.  155. 

N.B.   The  first  circle  must  be  drawn  sufficiently  large,  so 
that  the  second  circle  may  meet  it. 


241.    Problem  VI. —  To  construct  a  A  equal  to  a  given 
/Inside. 


CONS  TR  UC  TIONS.  195 

Solution.  Drawn  either  diagonal,  as  AC,  of  the  4-side 
ABCD,  and  then  from  D  draw  a  II  to  the  diagonal,  cutting 
AB  at  A\     Then  ACE  is  the  A  sought  (why?)  (Fig.  156). 


242.  Problem  VII.  —  To  construct  a  A  equal  to  a  given 
n-side. 

Proceed  as  in  Problem  VI,  and  reduce  one  by  one  the 
number  of  sides  drawn  to  three. 

243.  Problem  VIII.  —  To  divide  a  parallelogram  into  n 
equal  parts  by  parallels  to  a  side. 

Solution.  Divide  an  adjacent  side  into  n  equal  parts  and 
draw  parallels ;  the  n  resulting  parallelograms  are  congruent 
(why?). 

244.  Problem  IX.  - —  To  divide  a  A  into  n  equal  parts  by 
tracts  drawn  from  a  vertex. 

245.  Problem  X.  —  To  divide  a  trapezoid  into  n  equal 
parts  by  tracts  between  and  parallel  to  the  parallels. 

246.  Problem  XI.  —  To  divide  a  A  into  n  equal  parts  by 
tracts  drawn  from  a  point  on  a  side. 

Solution.  Divide  the  side  containing  the  point  into  n  equal 
parts ;  from  the  points  of  division  draw  parallels  to  the  join 
of  the  point  with  the  opposite  vertex ;  draw  tracts  from  the 


196 


GEOMETRY. 


point  to  the  intersections  of  these  parallels  with  the  other 
sides  ;   they  are  the  dividing  lines  required  (why?). 

247.    Problem  XII.  —  From  a  point  within  a  A  to  bisect  the 
A  by  tracts  drawn  to  a  given  vertex  and  to  a  side  (Fig.  157). 

C 


M 

Fig.  157. 

Solution.  If  P  be  the  given  point,  C  the  given  vertex, 
and  PQ  the  required  tract,  then  on  drawing  the  medial 
CM  it  becomes  plain  that  A  CPQ  =  A  CMQ ;  hence  CQ 
is  II  to  PM.  Hence  the  construction  :  draw  the  medial 
CM,  then  PM,  then  CQ  II  to  PM,  then  PQ. 

248.    Problem  XIII.  —  Prom  a  point  within  a  A  to  bisect 
the  A  by  two  tracts,  one  of  which  is  drawn  to  a  given  point 
on  one  side,  and  the  other  as  may  be  (Fig.  158). 
C 


CONS  TR  UC  TIONS.  197 

Solution.  If  P  be  the  given  point  within  the  A,  Q  the 
given  point  on  the  side,  suppose  PR  to  be  the  required 
tract.  Then  on  drawing  CM  and  PM  and  a  II  to  PM 
through  Q  cutting  CM  at  /,  we  have 

A  PQM=  A  PIM  (why?). 

Also  on  drawing  CP  and  IR  we  must  have 

APIC-APRC{^hy}). 

Hence  IR  is  II  to  PC  (why?). 
Hence  we  construct  PR  (how?). 

249.  Problem  XIV.  —  To  bisect  an  n-side  by  a  tract 
drawn  from  a  given  vertex. 

Solution.  Let  A  be  the  given  vertex ;  join  the  adjacent 
vertices  B  and  Z,  and  through  each  of  the  others  draw  a 
tract  across  the  «-side  II  to  BL.  Bisect  these  parallels  by 
a  train  of  tracts  from  A.  This  broken  line  will  bisect  the 
«-side  (why?),  and  by  Problem  VH  the  student  may  con- 
vert it  into  a  single  tract  from  A  (how?). 

250.  Problem  XV.  —  To  construct  a  square  equal  to  the 
sum  of  two  given  squares. 

Use  the  Pythagorean  Theorem. 

251.  Problem  XVI.  —  To  construct  a  square  equal  to  the 
sum  of  tivo  given  rectangles. 

Combine  the  methods  of  Problems  IV  and  XV. 

252.  Problem  XVII.  —  To  construct  a  square  equal  to  2, 
3,  4,  •  •  •  «  times  a  given  square  (Fig.  159). 

Hint.  The  diagonal  of  the  given  square  will  be  the  side 
of  the  double  square  (why  ?)  ;  normal  to  this  diagonal,  OB^ 
lay  of[  BC  equal  to  the  side  of  the  square  ;  draw  OC,  and 


198  GEOMETRY.  [Th.  CXXX^ 

again  normal  to  it  lay  off  CD  equal  to  the  side  of  the  origi- 
nal square ;  draw  OD,  and  so  on.  In  this  way  we  may 
duplicate,  triplicate,  ;2-plicate  the  original  square.  The 
broken  Hne  ABCD  •  •  •  and  the  varying  hypotenuse  wind 
round  O  forever. 

D 


253.  Problem  XVIII.  —  To  construct  a  square  equal  to 
one  half,  one  third,  one  fourth,  •  •  •  one  nth  of  a  give 71  square. 

Hint.  Half  the  diagonal  of  the  given  square  is  the  first 
side  sought ;  the  altitude  of  a  regular  triangle  whose  side  is 
the  given  side  is  the  second ;  one  half  of  the  given  side  is 
the  third ;  •  •  •  in  general,  the  geometric  mean  between  the 
side  and  the  ^th  part  of  the  side  of  the  given  square  will  be 
the  side  of  the  square  sought  (why  ?) . 

254.  Problem  XIX.  —  To  construct  a  square  the  Ti\h-fold 
or  the  nth.  part  of  a  given  i^ectangle,  parallelogram,  or  A. 

Combine  the  methods  of  the  foregoing  problems. 

255.  Theorem  CXXX^.  (Lemma).  —  If  a  jrc tangle  equal 
a  square,  and  the  dimensions  of  the  two  be  changed  propor- 
tionally {i.e:  so  that  the  new  and  the  old  dimensions  taken 
in  pairs  of  correspondents  form  a  continued  proportion), 
then  the  new  rectangle  will  equal  the  new  square. 


CONSTRUCTIONS.  199 

Data :  R  and  R\  two  rectangles  with  dimensions  a  and 
b,  a}  and  b\  s  and  s\  two  squares  with  dimensions  t  and  f ; 
R  =  S,2Lnda:a'::b'.b':'.f:f'  (Fig.  i6o). 


Fig.  i6o. 

Proof.  On  the  same  half-ray  from  the  same  point  P  lay 
off  tracts  FA  and  RB  equal  to  a  and  b ;  on  their  difference 
(AB  =  2  r)  as  diameter  describe  a  circle  and  draw  the  tan- 
gent PT:  it  will  equal  /  (why?).  With  a  radius  r'  such  that 
a  :  a' :  :  r:  r'  describe  a  concentric  circle,  prolong  OT  to 
meet  this  circle  at  T',  and  at  T'  draw  a  tangent  meeting 
the  diametral  ray  at  P. 

Then       PA'  =  «',  F'B'  =  ^',  /^/'  =  /  (why?), 
and  a'b'  =  /'/  (why  ?) .  q.  e.  d. 

Corollary.  If  two  rectangles  (or  parallelograms  or  A) 
be  equal,  and  their  dimensions  be  changed  proportionally, 
they  will  remain  equal. 

Prove  this  corollary  in  detail,  and  state  it  along  with  the 
foregoing  theorem  in  symbols. 

256.  Problem  XX.  —  To  construct  a  rectangle  similar  to 
a  given  rectangle  but  of  double  the  area. 

Solution.  Construct  a  square  equal  to  the  given  rectangle  ; 
then  either  dimension  of  the  double  rectangle  will  be  a 


200  GEOMETRY. 

fourth  proportional  to  the  side  and  diagonal  of  the  square 
and  the  corresponding  dimension  of  the  given  rectangle ; 
that  is, 

s  :  d:  :  a  :  a^  (why?). 

Now,  however,  all  squares  are  similar  (why?)  ;  hence, 
denoting  by  d'  the  diagonal  of  the  square  on  the  side  a,  we 
have  s  :  d\  '.  a  '.  d^ ;  whence  a  \  d^  \\  a  \  a\  ox  a^  ^=  d\ 

Similarly,  ^'  is  the  diagonal  of  the  square  on  the  other 
dimension  b.  Or  we  may  find  h^  by  drawing  a  diagonal  of 
the  double  rectangle  through  the  end  of  a^  parallel  to  the 
diagonal  through  the  end  of  a. 

257.  Problem  XXI.  —  To  construct  a  rectangle  similar  to 
a  given  rectangle  but  of  n-fold  the  area. 

Solution.  If  we  construct  a  square,  of  side  s,  equal  to  the 
given  rectangle  and  also  a  square,  of  side  j-',  equal  to  the. re- 
quired rectangle,  then  if  a  and  a^  be  corresponding  dimen- 
sions in  the  two  rectangles,  we  have 

s:  aw  s^  :  a^  (why?). 

Now,  however,  if  we  construct,  according  to  Problem  XI, 
two  broken  lines,  one  on  j-  as  a  basis,  the  other  on  a  as  basis, 
the  two  will  be  similar  figures  (why  ?)  ;  so  that  if  j"„  and  ^„  be 
corresponding  hypotenuses  in  the  two  figures,  we  have 


Hence  i-' :  ^' :  :  s^^ :  a^ ; 

hence,  if  s^  =  s\  then  a^^  =  a\ 

Accordingly,  we  find  «'  by  constructing  (Problem  XI)  the 
side  of  a  square  the  n-{o\A  of  the  square  on  a.  The  con- 
struction is  then  completed  (how?). 


CONSTRUCTIONS.  201 

258.  Problem  XXII.  — Let  the  student  extend  the  same 
methods  to  the  co7istruction  of  parallelograms  and  3\  similar 
to  giveti  ones  but  of  71-fold  area. 

259.  Problem  XXIII.  —  To  construct  a  rectangle  {paral- 
lelogram or  A)  similar  to  a  given  one  but  of  one  half,  one 
third  •  •  •  one  n\h  the  area. 

260.  Problem  XXIV.  —  To  construct  a  figure  similar  to  a 
given  figure  bu^t  of  double,  triple,  •  •  •  n-f old  area. 

Solution.  On  any  tract  in  the  figure  construct  a  square, 
then  construct  another  square,  of  double,  triple,  •  •  •  /z-fold 
area ;  its  side  will  be  the  tract  in  the  new  figure  corresponding 
to  the  assumed  tract  in  the  original  figure  (why?).  All  other 
points  and  lines  in  the  required  figure  may  now  be  found  by 
drawing  parallels.    Let  the  student  carry  out  the  construction. 

261.  Problem  XXV.  —  To  construct  a  figure  similar  to  a 
given  figuj-e  but  of  one  half,  one  third,  •  •  •  one  nth.  the  area. 

The  solution  is  like  that  of  Problem  XVIII,  mutatis  mu- 
tandis. 

262.  We  have  learned  to  inscribe  in  a  circle  a  regular 
3-side,  6-side,  i2-side,  •  •  •  3'2"-side,  also  a  regular  2"-side, 
and  it  is  natural  to  inquire  how  to  inscribe  a  regular  5 -side, 
lo-side,  •  •  • ,  5-2^-side.  As  it  was  easiest  to  begin  with  the 
6-side,  so  it  is  easiest  to  begin  with  the  lo-side ;  to  inscribe 
the  5 -side  directly  presents  difficulties. 

263.  Problem  XXVI.  —  To  inscribe  a  regular  10-side  in  a 
circle,  suppose  the  problem  solved  and  AB  the  side  sought. 
Then  in  the  symmetric  A  A  OB  the  vertical  ^  is  half  of 
either  basal  angle  (why?)  (Fig.  161). 

Hence,  if  we  draw  AC  bisecting  angle  A  the  A  AOB 
and  ^^C  will  be  similar  (why?). 


202 


GEOMETRY. 


Hence  OB  .  AB  :  :  AB  \  BC,  ox  OB  :  OC :  :  OC :  BC. 

Hence,  in  order  to  find  AB  or  OC,  it  is  necessary  to 
divide  the  radius  into  two  parts  of  which  one  is  the  geometric 
mean  of  the  whole  and  the  other.     This  celebrated  section  is 


called  the  median  or  golden  section,  and  the  radius  (or  any 
tract  so  divided)  is  said  to  be  divided  in  extreme  and  mean 
ratio. 

The  problem  of  inscribing  the  lo-side  is  reduced  then  to 
the  following : 

264.    Problem  XXVII.  —  To  divide  a   tract  in  extreme 
and  mean  ratio. 

Solution.    Let  a  be  the  tract,  and  b  the  greater  part ; 
then  a\b\\b\a  —  b  (Fig.  162), 

or  a\  a  -{-  b  \  \  b  :  a  (why  make  this  change  ?) . 


Fig.  162. 


CONSTRUCTIONS.  203 

Hence  we  may  conceive  of  a-  as  the  power  of  a  point 
whose  distances  from  the  circle  (along  a  diameter)  are  a-\-  b 
and  b  (why?),  so  that  a  is  the  diameter  of  the  circle.  Hence, 
on  a  as  diameter  draw  a  circle  ;  at  any  point  Z"  of  the  circle 
draw  a  tangent  and  on  it  take  TP=  a ;  draw  the  diameter 
PBA;  then  FB  =  b  (why?),  and  the  arc  about  F  with 
radius  b  divides  FT  or  a  at  I  in  extreme  and  mean  ratio. 

Q.  E.  F. 

N.B.  To  the  point  /corresponds  the  harmonic  conjugate 
O,  the  point  of  ou/er  median  section,  such  that 

F/:/T::FO:  OT. 

Let  the  student  show  that 

FT-  TO  =  pa,  just  as  FT-  TI^  F~l\ 

How  shall  we  now  construct  a  regular  5 -side,  20-side,  •  •  • 
5-2"-side? 

By  combining  the  constructions  for  a  3-side  and  a  5 -side 
we  may  now  construct  a  regular  15-side.  For  the  difference- 
of  the  arcs  subtended  by  a  side  of  a  regular  3-side  and  a 
side  of  a  regular  5-side,  is  (|^  —  1^)  of  a  circle,  or  ^-^  of  a 
circle ;  half  of  it  is  y^^,  or  (J  —  -^-^)  of  a  circle,  that  is,  the 
arc  subtended  by  one  side  of  a  regular  15-side.  Hence 
solve 

Problem  XXVIII.  —  To  cojistruct  a  regular  \^'2'^-side. 

265.  At  this  point  the  query  seems  to  arise  naturally :  if 
we  can  find  the  arc  of  a  side  of  a  15-side  by  combining  those 
of  a  3-side  and  a  5-side,  may  we  not  find  arcs  of  sides 
of  other  regular  ;/-sides  by  other  combinations?  To  take 
the  most  general  case,  let  us  form  the  difference  of  /  arcs 
of  a  2''«3-side  and  q  arcs  of  a   2'-5-side;    it  will   be    the 

/~^  '^  )th  of  a  full  angle;  i.e.   it  will  be    (2-5  — 2-3) 


204  GEOMETRY. 

times  the  arc  of  one  side  of  a  i5-2''  +  *-side ;  but  this  latter 
polygon  may  be  constructed  by  the  preceding  problem. 
Hence  nothing  new  is  obtained  by  the  new  combination. 
Herewith,  then,  the  round  of  elementary  construction  of 
regular  polygons  is  practically  completed  in  the  four  series : 
2'*-sides,  3-2"-sides,  5-2"-sides,  i5-2"-sides.  The  profound 
analysis  of  Gauss  has  indeed  shown  that  ruler  and  compasses 
will  suffice  to  construct  a  (2'*+ 1) -side  whenever  (2" 4-1)  is 
a  prime  number ;  and  accordingly  we  can  construct  regu- 
lar 17-sides  (;z  =  4)  and  2  5  7-sides  (;/ =  8)  ;  but  the  con- 
struction of  the  former  is  exceedingly  tedious,  and  that  of 
the  latter  is  excessively  so,  while  for  still  higher  values  of  n 
the  tedium  and  difficulty  surpass  all  limit.  However,  in 
figures  94,  95  a  regular  7-side  and  a  regular  9-side  are  con- 
structed once  for  all,  empirically,  but  to  practical  perfection. 

266.  Problem  XXIX.  —  To  draw  a  circle  through  two 
given  points,  tangent  to  a  given  ray. 

Hint.  Consider  the  power  of  the  intersection  of  the  given 
ray  and  the  ray  through  the  points  with  respect  to  the 
required  circle,  and  use  Problem  HI. 

267.  Problem  XXX.  —  To  draiv  a  circle  through  a  given 
point  and  tangent  to  tivo  given  rays. 

Hint.  Find  a  second  point  on  the  circle  and  apply  Prob- 
lem XXIX. 

268.  The  doctrine  of  perspective  similarity  may  often  be 
used  in  constructions. 

A.  When  one  datu7n  is  a  tract,  the  other  data  being 
angular  and  proportional  relations.  We  then  construct  in 
accordance  with  these  latter,  disregarding  the  first  one ;  in 
the  constructed  figure  a  tract  will  correspond  to  the  given 


CONSTR  UCTIONS. 


205 


tract,  and  on  this  latter  we  then  construct  the  required  figure 
similar  to  the  one  first  constructed. 

269.    Problem  XXXI.  —  Given  the  angles  and  an  altitude 
of  a  A,  to  construct  it  (Fig.  163). 


Fig.  163. 

Solution.  Draw  any  A  with  the  given  angles  ;  then  firom 
the  proper  vertex,  C,  lay  off  the  given  altitude  as  CD  normal 
to  the  opposite  side  AB.  Draw  through  Z)  a  II  to  AB  cut- 
ting the  other  side  at  A^B\  Then  A'B'C  is  the  required  A 
(why?).  The  two  A  ABC  and  A'B'C  are  perspectively 
similar,  C  being  the  centre  of  similitude. 

270.  B.  IVken  one  figure  is  to  be  inscribed  in  another  so 
that  certain  points  of  the  one  fall  on  certain  lines  of  the 
other,  we  may  draw  a  figure  in  perspective,  with  the  required 
figure,  as  to  the  intersection  of  two  rays  on  which  are  to  lie 
two  points,  and  then  from  this  centre  of  similitude  construct 
the  required  figure  according  to  the  remaining  conditions. 

271 .  Problem  XXXII.  —  To  inscribe  a  square  in  a  A  7vith 
the  vertices  of  the  square  on  the  sides  of  the  A  (Fig.  164). 

Solution.  Inscribe  any  square  HP  in  the  A,  and  draw 
^/^  meeting  BC  at  P' )  then  /^  is  a  point  of  the  required 
square.     Complete  the  construction. 


206 


272.  C.  It  is  often  required  to  inscribe  in  a  given  figure 
a  tract  that  shall  be  cut  by  a  given  point  proportionally  in  a 
given  ratio.  We  may  then  assume  the  given  point  as  centre 
of  simiHtude,  construct  a  figure  similar  to  the  given  figure 
with  the  given  ratio  of  simihtude ;  then  the  required  tract 
will  go  through  a  point  of  intersection  of  the  two  figures. 

273.  Problem  XXXIII.  —  To  draiv  through  a  point  I  in 
a  circle  S  {of  radius  r)  a  chord  that  shall  be  divided  by  I  in 
the  ratio  a  :  b  (Fig.  165). 


-.P 


\S' 


Solution.  From  /  lay  off  opposite  to  IC  the  tract  IC  so 
that  a  :  b  :  :  IC :  IC.  About  C  as  centre  with  radius  r', 
such  that  a\b\\r\r\  draw  a  circle  S  meeting  6"  at  M  and 
P.     Then  MN ox  FQ  is  the  chord  sought  (why?). 


CONS  TR  UC  TIONS. 


207 


How  will  you  proceed  in  case  of  outer  division?  The 
two  divisions,  inner  and  outer,  may  be  conveniently  distin- 
guished by  prefixing  the  sign  —  to  the  smaller  term  of  the 
ratio ;  i.e.  to  the  tract  corresponding  to  the  tract  that  will  be 
wholly  without  the  given  tract  after  division. 

*274.  The  following  discussions  might  have  been  intro- 
duced much  earlier,  at  Miscellaneous  Applications,  but  for 
interrupting  the  course  of  thought. 

We  may  conceive  the  area  of  a  parallelogram  as  generated 
by  slipping  one  of  its  sides  along  the  other  two  parallel  sides. 
Plainly,  the  side  slipped  is  merely  slipped  or  pushed,  not 
turned  at  all,  being  kept  parallel  to  itself  (as  the  phrase  is) 
throughout.  Thus,  suppose  the  tract  AB  slipped  along  the 
II  and  equal  tracts  AD  and  ^C;  it  will  generate  the  paral- 
lelogram area  ABCD. 

Clearly,  the  tract  may  be  slipped  along  the  same  parallels 
in  either  of  two  opposite  senses,  as  from  A  \.o  D  ox  from  D 
to  A.     The  sense  of  the  motion  of  the  tract  will  be  the  same 


Fig.  i66. 


208  GEOMETRY.  [Th.  CXXX. 

as  the  sense  of  the  motion  of  any  point  of  its  ray,  as  of  /, 
the  intersection  of  the  ray  with  any  other  ray,  as  with  the 
normal  ray  L  (Fig.  i66).  The  two  senses  of  /'s  motion 
may  be  distinguished  as  positive  and  negative  ;  then  the  cor- 
responding areas  generated  by  the  moving  tract  may  also 
be  distinguished  as  positive  and  negative.  In  summing  such 
areas  we  always  regard  the  sense  and  remember  that  to  add 
resp.  subtract  a  magnitude  is  the  same  as  to  subtract  resp. 
add  the  counter  magnitude.,  i.e.  the  magnitude  equal  in  size 
but  opposite  in  sense.  Bearing  this  in  mind  we  may  now 
enounce : 

*275.    Theorem  CXXX.  —  The  suin  of  the  areas  generated 
in  simply  slipping  a  tract  round  a  A  is  o. 

Data:  ABC  the  A,  A  A'  the  tract  in  its  initial  position 
(Fig.  167). 

C 


Fig.  167. 

Proof.  Suppose  the  tract  to  compass  the  A  counter- 
clockwise ;  then  if  the  area  AB'  be  considered  positive,  the 
areas  BC  and  CA'  must  be  considered  negative  (why?). 
But  on  taking  away  the  A  A'B'C  from  the  whole  figure  AB 
B'CA'  there  is  left  AB' ;  and  on  taking  away  ABC  there 


Th.  CXXXIL]  constructions.  209 

is  left  the  sum  of  BO  and  CA^ ;  hence  the  areas  AB^  and 
BC  -{■  CA'  are  equal  in  size  but  opposite  in  sense ;  hence 
their  sum  is  o,  or  AB'  -\-  B C  -\- CA'  =  o.     q.  e.  d. 

Corollary.  If  the  A  ABC  be  curvilinear  instead  of  recti- 
linear, the  theorem  still  holds. 

*276.  Theorem  CXXXI. — If  a  tract  be  simply  pushed 
round  any  closed  figure,  the  sum  of  the  areas  generated  will 
/^^o  (Fig.  1 68). 


Fig.  i68. 

Proof.  The  figure  may  be  cut  up  into  a  number  of  A 
rectilinear  and  curvilinear.  The  sum  of  areas  generated  in 
compassing  each  A  is,  by  the  foregoing  theorem,  o ;  hence 
the  total  sum  of  areas  generated  is  o ;  but  each  dividing 
tract,  as  AC,  is  compassed  twice,  in  opposite  senses,  from 
C  to  ^  and  from  A  X.o  C ;  hence  the  sum  of  areas  gener- 
ated along  these  divisions  is  o ;  subtracting  which  we  have 
left  the  sum  of  areas  generated  along  the  outer  border  equal 
to  o.     Q.  E.  D. 

*277.  Theorem  CXXXII  (of  Pappus,  a.d.  300) .  —  A  par- 
allelogram 071  one  side  of  a  i\  ivhose  counter-vertex  lies 
between  two  parallel  sides  of  the  parallelogram,  equals  the 
sum  of  two  parallelograms,  on  the  other  sides,  whose  parallel 
sides  go  through  the  vertices  of  the  first  parallelogra^n. 


210 


GEOMETRY. 


[Th.  CXXXII. 


Proof.     Let  the  student  show  from  the  figure,  by  help  of 
Theorem  CXXX  that  AB'  =  BC  +  CA  (Fig.  169). 


Fig.  169. 

Corollary.  As  a  special  case,  let  the  student  prove  the 
Pythagorean  Theorem. 

278.  Def.  The  tract  from  a  fixed  point  to  a  variable 
(or  moving)  point  is  called  the  radius  vector  of  the  moving 
point  with  respect  to  the  fixed  point.  Thus  OP  is  the 
radius  vector  as  to  O  of  the  point  P  as  it  traces  the  curve  C 
(Fig.  170). 


Fig.  170. 

Def.     The  area  bounded  by  the  path  of  the  moving  point 
and  two  positions  of  its  radius  vector  is  said  to  be  generated, 


Th.  CXXXIIL]  constructions.  211 

descri/?ed,  or  swept  out  by  the  radius  vector  in  passing  from 
the  initial  to  the  final  position.  Thus  the  area  POQ  is 
swept  out  by  r  in  passing  from  OF  to  position  OQ. 

Clearly,  the  same  area  may  be  described  in  either  of  two 
opposite  senses,  according  as  the  rotation  of  the  radius 
vector  is  clockwise  or  counter-clockwise,  and  the  area  must 
be  distinguished  accordingly. 

We  shall  call  areas  generated  clockwise  negative,  and 
areas  generated  counter-clockwise  positive.  Remembering 
the  laws  for  adding  and  subtracting  magnitudes  opposite  in 
sense,  we  now  enounce  : 

279.  Theorem  CXXXIII.  —  T/ie  total  area  generated  by  a 
radius  vector  whose  end  compasses  a  A  completely  is  the  A 
itself. 

Proof.  If  the  point  O  be  within  or  on  the  A,  the  validity 
of  the  theorem  is  immediately  evident.  If  the  point  O  be 
(Fig.  171)  without  the  A,  then  the  area  inside  is  generated 


but  once,  while  the  area  outside,  as  AOC,  is  generated  twice, 
in  opposite  senses  ;  once,  as  AOC,  clockwise,  once,  as  COA, 
counter-clockwise  :  such  will  always  be  the  case,  since  the 
final  and  initial  positions  of  the  radius  vector  are  the  same. 


212  GEOMETRY.  [Th.  CXXXIV. 

Hence  the  outside  areas  annul  each  other,  and  there  is  left 
only  the  inside  area,  the  A.     q.e.d. 

Corollary.    If  the  A  be  curvilinear,  the  theorem  still  holds. 

280.  Theorem  CXXXIV.  —  The  area  described  by  a  radius 
vector  whose  end  compasses  any  closed  figure  is  the  area  oj 
the  figure  itself. 

Proof.  Employ  the  method  and  reasoning  of  Theorem 
CXXXI.  Conduct  the  proof  carefully  in  the  case  of  a  ring 
and  of  a  loop.  Why  do  the  arrows  point  as  they  do  ?  What 
effect  will  reversing  one  have  on  the  other?  Imagine  the 
ring  slit  through  from  outer  to  inner  border  (Fig.  172). 


Fig,  172. 

The  foregoing  theorems  play  an  important  role  in  Higher 
Mathematics. 

THE   TACTION    PROBLEM. 

281.  In  the  following  discussions  certain  higher  concepts 
of  Geometry,  which  have  thus  far  been  lightly  passed  over 
or  not  formed  at  all,  become  regulative  and  must  receive 
graver  consideration.     We   begin   by   re-defining   some  of 


THE    TACTION  PROBLEM.  213 

them  and   recalling  some   of  their   already   demonstrated 
properties. 

1.  The  rectangle  of  the  distances  of  a  point  from  a  circle 
along  any  ray  through  the  point  is  called  the  power  of  the 
point  as  to  the  circle.  The  power  is  equal  to  the  square  on 
the  tan  gent- length  from  the  point  to  the  circle  when  the 
point  is  without,  and  equal  to  the  square  on  half  the  shortest 
chord  through  the  point  when  the  point  is  within  the  circle. 

2.  All  points  that  have  equal  powers  as  to  two  circles 
lie  on  a  ray  called  the  power-axis  of  the  two  circles.  The 
ray  is  normal  to  the  centre-tract  of  the  circles,  of  radii 
r  and  /,  and  divides  it  into  segments  d  and  d^  such  that 
{r+  r')-{r-  r')z={d+  d'){d-  d').  The  three  power- 
axes  of  three  circles,  taken  in  pairs,  concur  in  a  point 
called  the  power-centre  of  the  three  circles. 

3.  Any  two  points  P  and  /"  on  the  same  ray  through  a 
fixed  point  O  are  said  to  be  in  perspective  or  perspectively 
similar  as  to  the  centre  of  similitude  O  in  the  ratio  of 
similitude  0P\  0P\ 

4.  Two  figures  are  said  to  be  in  perspective  ox  perspectively 
similar  when  every  point  of  one  is  perspectively  similar  to 
the  corresponding  point  of  the  other  as  to  the  same  centre 
and  in  the  same  ratio  of  similitude. 

5.  When  OP  and  OP'  have  the  same  sense,  the  perspec- 
tive is  direct,  and  the  centre  outer ;  when  they  are  opposite 
in  sense,  the  perspective  is  counter,  and  the  centre  inner. 

6.  Any  two  circles  are  perspectively  similar  in  the  ratio  of 
their  radii  as  to  both  an  inner  and  an  outer  centre  ;  namely, 
the  points  dividing  the  centre  tract  harmonically  in  the  ratio 
of  the  radii,  which  are  also  the  points  of  intersection  of 
common  tangents  to  the  two  circles,  when  such  tangents 
there  are. 


214 


GEOMETR. 


[Th.  CXXXV. 


282.  Theorem  CXXXV.  —  Lemma.  —  If  two  figures  are 
similar  to  a  third,  they  are  similar  to  each  other. 

The  easy  proof  is  left  to  the  student. 

283.  Theorem  CXXXVI.  — If  07ie  figure  is  in  perspective 
with  each  of  two,  these  latter  are  in  perspective  with  each 
other,  and  the  three  centres  of  similitude  are  coUinear. 

Proof.  Let  P,  Q,  R  hQ  any  three  points  in  the  first  figure, 
F',  Q,  R^  and  P^\  Q",  R"  the  corresponding  points  in  the 
other  figures.  Then  RQR  and  P"Q"R"  (Fig.  173)  are 
similar  (why?),  and  FR",  QQ",  RR"  meet  in  a  point,  O' 


Fig.  173. 


th.  cxxxvil]    the  taction  problem.  215 

(why?),  as  to  which  they  are  in  perspective  (why?).  Like- 
wise PQR  and  P'QR'  are  similar,  and  PP\  QQ ,  RR' 
meet  in  a  point  6?". 

Now  let  PQ,  PQ,  /^"(2"  cut  a  O'  at  S,  S\  S",  and 
draw  SR,  SR',  SR".  Then  QRS,  Q'R'S',  Q"R"S"  are 
all  similar  (why?),  5  and  S'  are  in  perspective  as  to  O", 
S  and  S"  are  in  perspective  as  to  O',  and  hence  S'  and  S" 
are  in  perspective  as  to  O.  Hence  O  is  on  the  ray  O'O" 
(why?).     Q.E.D. 

Corollary.  Show  that  the  three  centres  of  similitude  are 
either  all  outer  or  else  one  outer  and  tivo  inner. 

Def.  If  a  point  bisect  every  chord  of  a  figure  drawn 
through  the  point,  it  is  called  the  centre  of  the  figure,  and 
the  figure  itself  is  said  to  be  centric. 

A  central  ray,  and  often  a  central  chord,  of  the  figure  is 
called  a  diameter. 

284.  Theorem  CXXXVII.  — If  tivo  similar  centric  figures 
be  in  perspective  as  to  one  pointy  -they  are  also  in  perspective 
as  to  a  second  point  (Fig.  174). 


216  GEOMETRY.  [Th.  CXXXVII. 

Proof.  Draw  two  diameters  of  the  one  figure  and  the 
two  corresponding  chords  of  the  other;  these  latter  will 
also  be  diameters  (why?),  and  the  centres  will  correspond. 
Join  the  ends  of  these  diameters  crosswise;  that  is,  the  end 
of  one  with  the  non-corresponding  end  of  the  other.  The 
intersection  of  these  two  cross-joins,  /,  is  a  second  centre  of 
similitude  (why  ?) .     q.  e.  d. 

Corollary  i.  Of  these  two  centres  of  similitude,  the  one 
is  outer,  the  other  inner ;  and  they  divide  the  centre  tract 
CC  harmonically. 

Corollary  2.  If  three  similar  centric  figures  be  in  per- 
spective they  have  six  centres  of  simiHtude,  and  of  these 
the  three  outer  are  coUinear,  as  are  also  any  one  outer  and 
the  two  other  inner. 

Def.  A  ray  on  which  lie  three  centres  of  similitude  is 
called  an  axis  of  similitude. 

Corollary.  There  are  four  such  axes,  one  outer  and  three 
inner. 

The  central  figure  with  which  we  have  especially  to  do  is 
the  circle. 

285.  Def.  When  the  rectangle  of  the  distances  from  a 
fixed  point  O  of  two  points,  F  and  P,  on  the  same  ray 
through  O,  is  constant,  the  two  points  are  said  to  be  inverse, 
or  in  inversion,  with  respect  to  O  as  centre  of  inversion. 

Def.  A  circle  about  the  centre  of  inversion,  with  the  side 
of  the  square  equal  to  the  rectangle  of  the  distances  for 
radius,  is  called  the  circle  of  inversion,  and  its  radius  the 
radius  of  inversion. 

Def.  If  while  one  of  the  inverse  points  as  F  describes  a 
curve  C  the  other  describes  a  curve  C,  then  C  and  C  are 
said  to  be  inverse  or  in  inversion  with  respect  to  O. 


Th.  CXXXVIIL]      THE    TACTION  PROBLEM. 


217 


*ej.  Let  a  ray  through  a  centre  of  similitude  of  two  cir- 
cles cut  each  in  a  pair  of  correspondent  points  P  and  Q^  P 
and  Q  ;  then  each  point  of  each  pair  has  a  correspondent  or 
homologous  point  in  the  other  pair,  as  P  and  P\  Q  and  Q ; 
also  each  point  of  each  pair  has  a  non-correspondent,  or 
contra-correspondent,  or  anti-homologous  point  in  the  other 
pair,  as  P  and  Q\  Q  and  P . 

286.  Theorem  CXXXVIII.  — Anti-homologous  points  of 
two  circles  are  ifiverse  with  respect  to  the  centre  of  similitude 
of  the  circles  (Fig.  175). 


Fig.  175. 

Proof.  Let  O  be  the  centre  of  simihtude  of  the  circles, 
CC  the  centre  ray. 

Then  2^  OAP=  ^  O'A'P  (why?)  and  ^  OAP=  ^  BQP 
(why?);  hence  :^  BQP=^  BA'P ;  hence  :^  BA'P  and 
^BQP  are  supplemental;  hence  BA'PQ  is  an  encyclic 
quadrangle  ;  hence  OQ  •  OP  =  OB  •  OA'.  Now  the  points 
O,  B,  A'  are  fixed ;  hence  the  rectangle  OB  •  OA'  is  con- 


218 


GEOMETRY. 


[Th.  CXXXIV« 


stant ;  hence  Q  and  P  are  inverse  as  to  O.    Similarly  prove 
that  /'and  Q  are  inverse,     q.  e.  d. 

Corollary.  OA  •  OB'  =  OT-  OT  ;  hence  the  radius  of 
inversion  about  (9  is  the  geometric  mean  of  (^T'and  OT', 
the  tangent-lengths  from  the  centre  of  similitude  to  the 
circles. 

287.  Theorem  CXXXIV".  —  The  inverse  of  a  circle  is  i Is  elf 
a  circle. 

Data :  In  the  figure  let  O  be  the  centre  of  inversion,  6" 
the  circle,  /  the  circle  of  inversion  with  radius  r. 

Proof.  Draw  6^7"  tangent  to  S  and  construct  OT^  so  that 
OT:r.'.r':  02\     Draw  a  normal  to  OT  at  T  meeting  OC 


Fig.  176. 


Th.  CXXXVP.]     the    taction  problem.  219 

at  C ;  with  radius  CV  draw  a  circle.  It  is  the  inverse 
sought.  For  it  is  the  circle  S  of  the  preceding  theorem 
(why?) J  in  which  /*and  Q,  ^and  P  were  inverse  as  to  O. 

288.  Theorem  CXXXV.  —  The  transverse  Joins  {chords) 
of  two  pairs  of  anti-homologous  points  of  two  circles  meet  on 
the  power-axis  of  the  circles. 

Data :  P  and  Q,  V  and  U\  two  pairs  of  anti-homologous 
points  in  S  and  S ;  PF  said  Q'U',  their  transverse  joins 
(chords)  (Fig.  176). 

Proof.  The  quadrangle  PFU'Q'  is  encyclic  (why?).  Let 
the  student  complete  the  proof. 

Def  If  a  circle  touch  two  other  circles,  the  ray  through 
the  points  of  touch  is  called  the  chord  (or,  better,  the  ray) 
of  Contact. 

289.  Theorem  CXXXVI''.  —  The  ray  of  contact  of  two 
circles  with  a  third  goes  through  a  centre  of  similitude  of  the 
two  circles  (Fig.  177). 

Proof.  For  a  point  of  contact  of  two  circles  is  a  centre 
of  similitude  of  the  two  (why?)  ;  hence  the  ray  of  contact 
goes  through  two  centres  of  similitudes ;  hence  it  is  an  axis 
of  simiHtude  and  goes  through  a  third  centre  of  similitude ; 
namely,  of  the  two  circles  (why?),     q.e.d. 

Corollary.  When  the  two  circles  are  touched  similarfyy 
both  innerly  or  both  outerly,  the  ray  of  contact  goes  through 
the  outer  centre  of  similitude  of  the  two ;  when  they  are 
touched  dissimilarly,  one  innerly  the  other  outerly,  the  ray 
of  contact  goes  through  the  imter  centre  of  similitude  of 
the  two  (why?). 

290.  Def  The  ray  through  one  of  two  inverse  points 
normal  to  their  junction-ray  is  called  the  polar  of  the  other 


220 


GEOMETRY. 


[Th.  CXXXVII*. 


point,  and  this  latter  point  is  called  the  pole  of  the  polar, 
with  respect  to  the  circle  of  inversion.  This  circle  we  may 
call  the  circle  of  reference,  or  the  referee-circle,  or  simply  the 
referee.  Note  carefully  that  pole  and  polar  have  no  mean- 
ing except  with  respect  to  some  referee. 


Fig.  177. 


291.  Theorem  CXXXVII".  — If  of  two  points  the  first  is 
on  the  polar  of  the  second,  then  the  seco?id  is  on  the  polar  of 
the  first. 

Data :  S  the  circle,  P  the  first  point  on  the  polar,  Z,  of 
the  second  point  Q  (Fig.  178). 

Proof.  Draw  the  centre-ray  OQ  cutting  L  at  Q,  then  Q 
is  the  inverse  of  Q  (why?)  ;  also,  let  P  be  the  inverse  of  P. 
Then   the   quadrangle  PPQQ   is    encyclic    (why?)  ;    also 


th.  cxxxvip.]    the  taction  problem. 


221 


the  ^  ^'  is  a  right  angle  (why?)  ;  hence  so  is  the  ^ /*' 
(why?)  ;  hence  P^ Q  is  the  polar  oi P  (why?),  i.e.  the  polar 
of /*  goes  through  5.     q.e.d. 


Fig.  178. 

Corollary  i.  The  poles  of  all  rays  through  P  lie  on  the 
polar  of/*;  in  other  words,  as  a  polar  turns  about  a  point, 
its  pole  glides  along  the  polar  of  that  point. 

Corollary  2.  The  polars  of  all  points  on  a  ray  pass  through 
the  pole  of  that  ray ;  in  other  words,  as  a  pole  glides  along 
a  ray,  its  polar  turns  about  the  pole  at  that  ray. 

Scholium.  By  definition  the  rectangle  OP-  OP^  is  con- 
stant in  area ;  hence  as  P  moves  in  towards  O,  P'  moves 
out,  and  with  it  the  polar  of/*;  as  /*  falls  on  6>,  P  and  the 
polar  of  P  through  it  move  out  and  vanish  in  infinity ;  as  P 
moves  out  from  O  leftward,  P  and  the  polar  reappear  in 
infinity  on  the  left,  approaching  S ,  2i?>  P  reaches  S,  so  does 
P,  and  the  polar  becomes  a  tangent.  Hence  we  may  define 
the  tangent  as  a  polar  whose  pole  is  on  it  (the  polar). 


222 


GEOMETRY. 


[Th.  CXXXVIIl' 


292.    Theorem  CXXXVIII'.  —  7;z;/^<f«^^  at  the  end  of  a 
chord  meet  07i  the  po/ar  of  the  chord  (Fig.  179). 


Fig.  179. 

Proof.  For  the  chord  goes  through  the  poles  of  the  tan- 
gents (where  ?)  ;  hence  the  tangents  go  through  the  pole  of 
the  chord  (why?). 

Corollary.  Tangents  at  the  end  of  a  chord  through  a 
point  meet  on  the  polar  of  the  point. 

Hence  we  may  define  the  polar  of  a  point  as  the  locus  of 
the  intersection  of  the  pair  of  tangents  at  the  ends  of  any  chord 
through  the  points. 

Exercise.  Show  how  to  construct  the  polar  of  a  point 
without,  within,  or  upon  the  circle  of  reference. 

Def.  Two  points,  each  on  the  polar  of  the  other,  and  two 
polars,  each  through  the  pole  of  the  other,  are  called  con- 
jugate. 


THE    TACTION  PROBLEM. 


223 


We  are  now  prepared  to  attack 

THE   TACTION    PROBLEM. 

To  draw  a  circle  tangent  to  three  given  circles. 

293.  Lemma  A.  —  The  power-centre  of  three  circles  is  a 
centre  of  similitude  of  two  circles  each  tangent  to  each  of  the 
three. 


Fig.  180. 

Data :  6*,,  6*2,  6*3,  the  three  circles,  O  and  (9'  two  circles 
touching  them,  O  outerly,  0  innerly  ;  j;,  7^,  7^,  T\,  T\_,  T^, 
the  points  of  touch. 

Proof.    Draw  the  rays  of  contact,  TxT\,  T^T^,  T^T'.^. 

The  first  goes  through  two  points  of  dissimilar  contact  of 


224  GEOMETRY. 

6"  with  O  and  (9' ;  hence  it  goes  through  the  inner  centre 
of  simiHtude  of  O  and  (9' ;  the  hke  may  be  said  of  T^V^  and 
T^T\)  hence  these  rays  concur  in  P,  the  inner  centre  of 
simiHtude  of  O  and  0\ 

Hence  T^  and  T\,  T,  and  T\,  T^  and  T'^  are  three  pairs 
of  anti-homologous  points  on  O  and  O'  with  respect  to  P; 
hence         PT,  •  PT\  =  PT^  •  PT\  =  PT^  •  PT\  ; 
that  is,        P  is  the  Power- centre  of  S^,  So,  S.^.     q.  e.d. 

294.  Lemma  B.  — An  axis  of  similitude  of  three  circles  is 
a  power-axis  of  two  circles  tangent  each  to  each  of  the  three. 

Data :    The  same  as  before. 

Proof.  The  transverse  joins  T^P^  and  T\T\_  meet  on  the 
power-axis  of  O  and  O^  (why?)  ;  so  too  the  transversals 
T^P^  and  V^T-,,  P.P^  and  T^T\.  But  P.P.  and  P\P\ 
are  rays  of  contact  of  O  with  S^S.,  and  of  O^  with  ^'i^'^ ; 
hence  they  meet  in  the  (outer)  centre  of  simiHtude  of  S^ 
and  6*2;  similarly  for  the  pairs  P.P;,  and  r'.r',,  7^7;  and 
P\P\.  Hence  the  outer  axis  of  similitude  of  6'i,  S<2,  S:^  is 
the  power-axis  of  O  and  0\     q.e.d. 

295.  Lemma  C.  —  Phe  ray  of  contact  of  each  circle  with 
the  tivo  circles  goes  through  the  pole  as  to  the  circle  of  an  axis 
of  similitude  of  the  three  circles. 

Data :    The  same  as  before. 

Proof.  The  tangents  J/2^2  and  MoP\  are  equal;  hence 
the  point  M^  is  on  the  power-axis  of  O  and  O^,  i.e.  on  the 
(outer)  axis  of  simiHtude  of  ^1,  S^,  S^.  So,  too,  for  M^,  and 
Ml,  which  latter  in  the  figure  lies  at  infinity.  But  Mo  is  the 
pole  as  to  So  of  the  contact-ray  7^2^'2  (why?)  ;  hence  the 
pole  of  the  contact-ray  lies  on  the  axis  of  similitude  ;  hence 
the  pole  of  the  axis  of  similitude  lies  on  the  contact-ray,  or 
the  contact-ray  goes  through  the  pole  of  the  axis  of  simili- 
tude.    *Q.  E.D. 


THE    TACTION  PROBLEM.  225 

296.  Accordingly,  we  know  two  points  of  each  contact-ray, 
namely,  the  power-cetitre  of  the  three  circles  and  the  pole  of 
an  axis  of  similitude  as  to  each  circle.  We  have  then  this 
rule  of  construction  : 

Solution,  (i)  Find  the  power-centre  and  an  axis  of 
similitude  of  the  three  circles;  (2)  find  the  pole  of  this 
axis  as  to  each  of  the  circles ;  from  the  power-centre  draw 
three  rays  through  the  three  poles  :  they  cut  the  three  cir- 
cles in  three  pairs  of  points,  namely,  the  points  of  tangency 
of  two  required  circles. 

Thus  it  appears  that  each  axis  of  similitude  yields  in 
general  two  tangent  circles ;  and  there  are  four  such  axes ; 
hence  there  are  in  general  eight  tangent  circles. 

The  kind  of  tangency  is  determined  by  the  axis  of  simili- 
tude :  if  this  be  outer,  then  each  of  the  two  circles  touches 
all  three  similarly,  one  outerly,  the  other  innerly ;  if  the  axis 
be  inner,  but  drawn  through  the  outer  centre  (say)  of  S^ 
and  S,,  then  one  of  the  circles  will  touch  Sy  and  6*2  outerly, 
but  ^3  innerly,  while  the  other  will  touch  S^  and  ^'2  innerly, 
but  ^3  outerly. 

297.  This  classic  problem,  in  which  the  elementary 
geometry  of  the  circle  seems  to  culminate,  was  proposed 
and  solved  by  Apollonius  of  Pergae,  a.d.  200.  His  solution 
was  indirect,  reducing  the  problem  to  ever  simpler  and 
simpler  problems.  It  was  lost  for  centuries,  but  was  restored 
by  Vieta.  The  direct  solution  similar  to  the  foregoing  was 
first  given  by  Gergonne  (1813).  The  analogous  problem 
for  space,  namely,  to  construct  a  sphere  tangent  to  four 
given  spheres,  was  first  solved  by  Fermat  (1601-1665). 

The  foregoing  construction  is  immediately  applicable  to 
this  problem,  on  changing  3  into  4  and  ray  into  plane. 

It  is  important  that  the  student  actually  carry  out  the 
preceding  solution. 


226 


GEOMETRY. 


METRIC    GEOMETRY. 

298.  Thus  far  our  treatment  of  the  subject  of  Geometry 
has  been  strictly  geometrical ;  we  have  at  no  point  invoked 
the  aid  of  number,  Arithmetic,  or  Algebra  in  demonstration, 
so  that  if  these  sciences  should  suddenly  vanish  from  cogni- 
tion the  structure  of  our  geometric  knowledge  would  remain 
wholly  unimpaired.  Nevertheless,  in  the  Art  of  Geometry, 
in  the  practical  application  of  the  science  to  quantitative 
problems,  it  becomes  highly  important  to  express  linear, 
angular,  and  areal  magnitudes  through  numbers  or  at  least 
numerically,  and  to  apply  to  such  expressions  the  laws  of 
numerical  calculus.  Such  is  the  subject  of  the  following 
sections. 

299.  Def.  A  geometric  magnitude  (tract,  angle,  area)  that 
may  be  regarded  as  the  sum,  or  that  equals  the  sum,  of  m 
equal  geometric  magnitudes  (of  the  same  kind)  is  called  a 


B 

Fig.  181. 


multiple  (more  precisely,  the  m-fold)  of  one  of  those  equal 
magnitudes.  Thus  B  is  the  double  of  A,  D  the  triple  of  C, 
/^the  four-fold  of  ^  (Figs.  181,  182). 


Th.  CXXXIX.]         metric   geometry.  227 

Def.  Any  one  of  in  equal  geometrical  magnitudes  is  called 
an  mth  part,  or  simply  an  mth,  of  the  sum  or  of  the  equal 
of  the  sum  of  these  ;;/  magnitudes. 

Thus  ^  is  a  half  of  j9,  C  is  a  third  of  />,  ^  is  a  fourth 
(part)  oi  F. 

The  symbol  for  the  w-fold  of  a  magnitude  is  formed  by 
prefixing  m  to  the  symbol  for  the  magnitude.  Thus,  2  A^ 
3C,  40  ,  f7iM.  The  symbol  for  an  ;;^th  part  of  a  magnitude 
is  commonly  formed  by  writing  m  below  the  magnitude  and 
separating  the  two  by  a  horizontal  or  oblique  bar  :  thus, 

^,  ^,   n/4,  Qlm. 
2      3 

300.  Theorem  CXXXIX.  —  The  p-fold  of  the  mth  part  of 
a  magnitude  equals  the  mth  part  of  the  p-fold  of  the  magni- 
tude. 

Proof.  Let  Q  be  any  magnitude  (tract,  angle,  area).  By 
definition  there  are  m  ;;/th  parts  of  it.  In  its  /-fold  each 
such  part  will  be  present  /  times ;  hence  there  will  be  pm 
m\\v  parts  in  the /-fold  of  Q. 

Also  by  definition  the  ;;/th  part  of  this  /-fold  taken  m 
times  in  summation  must  yield  the  whole.  Now,  however, 
if  we  take  /  of  the  wth  parts  of  Q  and  take  them  m  times 
in  summation,  we  shall  get  a  whole  consisting  of  mpmih.  parts. 
But  it  is  a  fundamental  law  of  counting,  called  the  Commu- 
tative Law  of  Multiplication,  that  to  count  in  p  times  yields 
the  same  number  as  to  count  /  ;//  times.  Hence  this  whole 
is  equal  to  the  /-fold  of  Q ;  and  its  m\h  part  consists  of  / 
mi\i  parts  of  Q,  or  is  the  /-fold  of  the  mth.  part  of  Q]  i.e. 
the  /-fold  of  the  mth.  part  of  Q  equals  the  ;//th  part  of  the 
/-fold  of  Q.     Q.  E.  D, 


228  GEOMETRY.  [Th.  CXXXIX. 

Scholium.   The  /-fold  of  the  wth  part,  or  the  ;«th  part  of 
the  /-fold,  of  Q  is  commonly  written 

PQ      P  ^  Q 

— ,  or    '  'Q  ox  p  '  — 
m^        m    ^        ^     m 

The  expression   t.,  or  p/ut  is  called  a  fraction,  /  and  w 

its  terms,  p  the  numef-ator,  f?t  the  denominator.     We  have 
just  learned  what  it  means. 


301.  If  now  we  conceive  any  whole,  w,  as  the  sum  of  m 
equal  parts,  each  equal  to  u,  we  may  call  u  the  unit  magni- 
tude or  magnitudinal  unit.  Thus  one  yard  is  a  linear,  one 
degree  an  angular,  one  acre  an  areal,  unit.  There  may  be 
several  other  magnitudes,  the  /-fold,  ^-fold,  jc-fold  of  this 
same  unit  u.  Then  w,  /,  q,  x  are  called  the  metric  numbers 
of  these  magnitudes. 

302.  It  may  happen  that  a  magnitude  may  not  be  com- 
posable  out  of  equal  units  u ;  it  may  not  be  a  multiple  of 
the  unit-magnitude  u,  but  may  be  greater  than  the  /-fold  of 
u  and  less  than  the  (/ 4-  i)-fold  of  11.  Thus  a  circle  is 
more  than  triple,  yet  less  than  quadruple,  its  diameter.  In 
such  cases  it  may  be  possible  to  find  some  smaller  unit  of 
which  the  unit  u  is  the  w-fold,  and  the  other  magnitude 
(say)  the  ^-fold.  Thus  the  table  may  be  more  than  3  feet 
and  less  than  4  feet  long ;  but  on  changing  the  linear  unit 
from  the  foot  to  the  inch,  the  twelfth  of  a  foot,  we  may  find 
that  the  length  in  question  is  precisely  the  40-fold  of  the 
new  unit  —  the  table  is  precisely  40  inches  long.  Then 
40  is  the  metric  number  of  the  table-length  in  inches  and 
the  fraction  ^.o.  is  the  metric  number  of  the  same  length  in 
feet,  which  means  that  the  sum  of  40  12th  parts  of  a  foot  is 
the  length  of  the  table. 


th.  cxl.j  metric  geometry.  ll") 

303.  Often,  however,  in  fact  generally,  it  will  be  impossi- 
ble to  find  any  unit- magnitude  so  small  that  its  w-fold  shall 
be  the  one  magnitude  and  its /-fold  the  other;  and  this  im- 
possibility may  be  objective,  not  subjective  —  it  may  inhere 
in  the  nature  of  the  case  and  not  arise  from  some  defect  of 
our  own  powers  of  measurement  or  calculation.  Thus,  there 
is  no  unit-length,  however  small,  out  of  which  may  be  com- 
posed both  the  side  and  the  diagonal  of  a  square  ;  there  is 
no  length  so  small  that  the  side  shall  be  its  w-fold  and  the 
diagonal  its  ^-fold.  This  important  fact  may  be  estabhshed 
thus  : 

304.  Lemma.  —  If  each  of  two  tracts  is  a  multiple  of  the 
same  tract,  the  difference  of  the  two  is  a  multiple  of  the  same 
tract. 

Proof.  Let  G  be  the  greater  and  L  the  less  of  the  two 
tracts.  Then  we  have  G=pt  and  L^q- 1;  the  difference  of 
these  two  is  (/  —  q)t,  and  this  is  a  multiple  of  /,  since  the 
difference  of  two  integers,/  and  q,  is  itself  an  integer,/—^. 

305.  Theorem  CXL.  —  The  side  and  diagonal  of  a  square 
are  incommensurable  (Fig.  182). 

Proof.  Let  A^Bi  =  Si  be  the  side,  and  A1A2  =  di  be  the 
diagonal  of  a  square.  On  A1A2  lay  off  A^B.,  =  Si ;  then  A2B2 
is  the  difference  of  the  side  and  diagonal  and  is  therefore  a 
multiple  of  any  tract  of  which  Si  and  di  are  multiples  ;  call  it 
.f2.  Draw  ^0^3  normal  to  A1A2 ;  then  B^A^  =  ^^2^3  (why?) 
=  A2B2  (why  ?) .  Hence  ^2^3  is  the  diagonal,  dz,  of  a  square 
whose  side  is  A^Bo  or  S2.  Also  ^2  is  a  multiple  of  any  tract 
of  which  i-]  and  S2  are  multiples.  Hence  we  have  a  new 
square  with  side  j-j,  and  diagonal  ^2>  both  multiples  of  any 
tract  of  which  s^  and  di  are  multiples.  Also  the  new  side 
and  diagonal  are  respectively  less  than  half  of  the  old  side 
and  diagonal.     By  repeating  this  process  we  obtain  a  third 


230 


GEOMETRY. 


[Th.  CXL. 


square  with  side  and  diagonal  less  than  half  the  side  and 
diagonal   of  the    second,   less    than    one    fourth   those   of 


Fig.  182. 

the  first.  By  such  constructions  we  shall  obtain  a  square 
with  side  and  diagonal  less  than  {— Vh  of  the  side  and 
diagonal  of  the  original  square  ;  i.e. 

2"  2^ 

By  making  n  large  enough  we  may  make  i"„  and  d^,  as  small 
as  we  please,  small  at  will,  —  we  may  in  fact  sink  them 
below  any  assigned  degree  of  parvitude.  Meanwhile  they 
must  remain  multiples  of  any  tract  of  which  s^  and  ^1  are 
multiples ;  but  they  can  be  multiples  only  of  a  tract  smaller 
than  themselves,  manifestly ;  hence  any  tract  of  which  both 
Sy  and  dx  are  multiples  must  be  smaller  than  a  tract  as  small 
as  we  please.  But  there  is  no  such  tract,  self-evidently. 
Hence  there  is  no  tract  of  which  both  diagonal  and  side  of 
a  square  are  multiples,     q.  e.  d. 


Th.  CXLI.]  metric   geometry.  231 

Magnitudes  that  are  thus  not  multiples  of  the  same  mag- 
nitude are  said  to  have  no  common  measure,  or  to  be  in- 
commensurable. 

306.  Now  suppose  the  side  s  cut  up  into  some  very  large 
number  of  equal  parts,  as  q ;  then  the  diagonal  ^  will  not  be 
the  sum  of  any  number  of  these  parts  but  will  be  more  than 
the  sum  of/  parts  and  less  than  the  sum  of  (/  +  i)  parts ; 

/                 /+  I 
that  is,  -  '  s  <  d<, •  s. 

q  ^      q 

Here  we  may  make  q  as  large  as  we  please  ;  hence,  if  we 
take  s  for  our  linear  unit,  we  may  shut  in  the  metric  number 

of  ^  between  two  fractions,  ^  and  ^lJL^,  that  differ  by  -, 

q         q  q 

that  is,  by  a  fraction  small  at  will,  while  the  metric  number 
of  d  differs  from  each  by  less  than  -.     In  this  last  sentence 

q 

we  have  subreptively  assumed  that  d  has  some  metric  number 
when  referred  to  s  as  unit-length.  But  we  shall  not  build 
anything  on  this  assumption  at  present.     See  Art.  256. 

307.  We  now  pass  to  the  metric  numbers  of  area.  We 
agree  once  for  all  that  the  square  on  a  linear  unit  shall  be 
an  areal  unit.  The  metric  number  of  an  area  will  then  be 
the  number  of  areal  units  of  which  it  is  composed,  or  its 
equal  is  composed. 

308.  Theorem  CXLI.  —  The  metric  number  of  a  rectan- 
gle is  the  product  of  the  metj'ic  numbers  of  its  dimensions. 
Two  cases  arise  : 

T.    When  the  dimensions  are  conunensurable  (Fig.  183). 
Let  a  and  b  be  the  dimensions  of  a  rectangle.     Choose 
any  unit  of  which  a  and  b  are  both  multiples,  as  u,  so  that 


232 


GEOMETRY. 


[Th.  CXLI. 


a=  p  •  u,  b  =  q  '  ti.  Then  we  may  cut  up  a  into/  parts,  and 
b  into  q  parts  equal  to  u.  Through  the  points  of  division 
draw  Hi"  to  the  sides.      Then  the  whole  rectangle  will  be  cut 


n 

h 

Fig.  183. 

up  into  pq  squares  each  on  the  side  u  (why?)  ;  hence  the 
rectangle  will  be  the  sum  of  these  areal  units ;  hence  the 
metric  number  is  the  product/^,     q.e.d. 

Now  suppose  we  choose  some  larger  areal  unit,  as  the 
square  not  on  u  but  on  7^u.  In  this  square  there  are,  by  the 
foregoing,  rr  units ;  that  is,  it  is  the  rr-fold  of  the  small  unit 


w ;  or  the  small  unit  is  the  (  -  Jth  part  of  the  large  areal 

unit ;  hence  the  rectangle,  which  is  the  pq-ioXd  of  the  small 

(pq\ 
unit,  is  the    —  th  of  the  large  areal  unit.      But  the  sides 

^"^      (P\  fA 

were  respectively  the  (  -  1th  and  the  (  -  jth  of  the  small  linear 

unit  u ;  and  the  product  of  these  two  fractions  is,  by  the 

Pq 
laws  f 07-  i7iultiplying  fractions,  — .     Hence,  if  we  call  frac- 
tions numbers,  we  must  have  the  metric  number  of  the  area  is 

pq 

^,  or  is  the  product  of  the  metric  numbers  of  the  dimen- 
sions.     Q.  E.  D. 

2.    Now  suppose  the  two  dimensions  incommensurable  with 
the  linear  unit  u.     Then  a  will  be  >  ///  and  <  (/  +  i )  // ; 


Th.  CXLL]  metric   geometry.  233 

while  b  will  be  ^  q  •  u  and  <  (^  +  i)?^.  Then  plainly  there 
are  pq  squares  on  //  in  the  rectangle,  with  some  remainder, 
but  not  {pq  -\-p  4-^+1)  squares  ;  or 

pq  .  ?/  <ab  <{pq-\-p  +  q-\-  i)u\ 

Hence,  if  the  rectangle  ab  has  any  metric  number  at  all,  the 
same  must  He  between  the  values/^  and  (/  +  i)-(^-l-  i)  ; 
i.e.  it  must  He  between  the  product  of  the  metric  numbers 
too  small  and  the  metric  numbers  too  great  for  the  dimen- 
sions. Now  u  was  very  small,  and  hence  p  and  q  very  large. 
Take  a  new  unit  U-=rr-  u^,  whose  side  is  the  rth  multiple 
of  the  side  of  the  same  square.     Then  the  metric  numbers 

P  Q  p  -\-  "^ 

of  the  dimensions  a  and  b  will  be  >  -  and  -  but  < 

r  r  r 

q  -{-  1 
and ,  and  the  metric  number  of  the  area,  if  it  have  a 

pq  (/+  i)  (^+1) 

metric  number,  will  be  >  —  but  < :  i.e.  it 

'  rr  rr  ' 

will  lie  between  these  two  fractions  and  differ  from  each  by 

/  +  $^+  I 
less  than .     Now  with  any  fixed  unit  length,  as  i, 

P  ^ 

we  may  find  two  fractions,   -  and  -,  that  differ  from  the 

metric  numbers  of  a  and  b  (if  they  have  any)  by  less  than  -> 

and  by  taking  r  even  greater  and  greater  we  may  approach 
p         q 

our  fractions  -  and  -  close  at  will  to  the  metric  numbers  of 
r         r 

p         q 
a  and  b.    Each  of  these  fractions  -  and  -  meanwhile  remains 

r         r 

less  than  some  assignable  whole  number ;  so,  too,  does  their 

p  -\-  q  p  -{-  q  -{-  1 

sum ,  and  so  does .     Now  the  difference  of 

r    '  r 

the    fractions  —   and  is  the   rth  part  of 

rr  rr  ^ 

,  and  by  making  r  large  at  will  we  may  make  this 


234  GEOMETRY.  [Th.  CXLII. 

rth  part  small  at  will.  Hence  the  two  fractions  may  be  brought 
as  close  together  in  value  as  we  please,  while  between  them 
lies  always  the  metric  number  of  the  area,  and  also  between 
them  lies  always  the  product  of  the  metric  numbers  of  the 
dimensions.  These  two  numerics,  then,  the  metric  number 
of  the  area  and  the  product  of  the  metric  numbers  of  the 
dimensions,  cannot  differ  by  any  assignable  value  however 
small,  since  they  both  He  between  two  values  which  may  be 
made  to  differ  by  less  than  any  assigned  value  however 
small.  Hence  we  conclude  (ist)  that  these  two  numerics 
are  Definites,  since  the  bounds  between  which  each  lies, 
and  which  close  down  together  upon  each  other,  are  at 
every  stage  perfectly  definite,  and  (2d)  that  they  are  abso- 
lutely the  same  in  value. 

309.  It  is  a  matter  not  of  logical  compulsion  but  of  con- 
venient choice  to  call  this  Definite  a  number  or  at  least  a 
numeric.  Since  it  is  not  expressible  as  a  fraction,  still  less 
an  integer,  it  is  commonly  called  an  Irrational.  The  laws 
of  operation  on  the  algebraic  symbols  of  such  Irrationals  as 
well  as  Fractions  are  not  matters  of  logical  proof,  but  of 
allowable  assumption.  It  is  convenient  to  assume  for  them, 
arbitrarily  to  impose  upon  them,  the  same  laws  of  operation 
that  are  found  empirically  to  hold  for  positive  integers,  or 
numbers  obtained  by  counting.  This  fact  is  sometimes 
called  the  Principle  of  the  Permanence  of  the  Formal  Laws 
of  Operation  (Hankel).  Further  discussion  of  the  subject 
belongs  to  Algebra  and  would  be  out  of  place  here. 

310.  Knowing  that  the  metric  number  of  a  rectangle  is 
the  product  of  the  metric  numbers  of  its  dimensions,  we 
now  declare  at  once  that 

Theorem  CXLII.  —  The  metric  number  of  a  parallelogram 
is  the  product  of  the  metric  numbers  of  its  dimensions. 


Th.  CXLIV.]  metric   geometry.  235 

Theorem  CXLIII.  —  The  metric  number  of  a  A  is  half  the 
product  of  the  metric  numbers  of  its  di?nensiojts. 

Theorem  CXLIV.  —  The  metric  nurtiber  of  a  trapezoid  is 
half  the  product  of  the  metric  numbers  of  its  altitude  and  the 
sum  of  its  II  bases. 

In  a  word,  all  the  theorems  that  declare  relations  among 
areas  may  now  be  translated  into  theorems  that  declare  like 
relations  among  the  metric  numbers  of  areas.  This  easy 
exercise  is  left  for  the  student. 


311.    We  have  thus  far  treated  proportion  strictly  geo- 
metrically.    We  have  written  off  the  symbolism 

a\b\\c\d, 

when  a^  b,  c,  d,  were  signs  for  tracts,  but  when  asked  what  we 

meant  by  it  our  only  reply  was,  we  mean  that  the  rectangle 

ad  equals  the  rectangle  be.     This   reply  was   perfect  and 

complete.     Now,  however,  if  instead  of  the  tracts  we  put 

p,  q,  r,  s,  as  the  metric  numbers  of  the  tracts,  we  may  still 

write  as  before 

p  :  q  :  '.  r  :  s, 

and  answer  the  question  what  this  means,  by  saying  it  means 
that  the  product  ps  equals  the  product  qr,  for  we  have  just 
proved  this  equaHty.  This  answer  is  also  perfect  and  com- 
plete.   However,  it  is  not  the  only  possible  answer.     For  we 

might  say  we  mean  that  the  fraction^  equals  the  fraction  -  ; 

q  s 

this  would  also  be  correct.  For  if  ps  =  qr,  then  on  divid- 
ing both  sides  by  qs  we  get  -  =  - ,  and  conversely,  if  ^  —  _, 

q      s  ^      ^ 

then  on  multiplying  both  sides  by  qs  we  get  ps  =  qr.  Ac- 
cordingly these  two  answers  are  equally  adequate  and  involve 


236  GEOMETRY. 

each  other.  But  we  could  not  make  any  such  second  answer 
to  the  question,  what  do  we  mean  by  the  proportion 

a:  b::  c  :  d} 

For  we  cannot  attach  any  meaning  to  the  symbolism  -  when 

b 

a  and  b  are  tracts,  nor  can  we  tell,  at  least  at  present, 
what  we  mean  by  dividing  one  tract  by  another.     We  may 

indeed  write  -  =  - ,  and  answer  the  inquiry  as  to  our  mean- 
b      d 

ing  by  saying  we  mean  the  rectangle  ad  equals  the  rectangle 
be ;  but  we  cannot  deduce  the  relation  reetangle  ad  =  rec- 
tangle be  from  the  symbolism  -  =  -  by  multiplying  through 

b      d 

by  the  rectangle  bd,  for  we  do  not  attach  any  meaning  to  the 
phrase  "  multiplying  by  a  rectangle." 

312.  The  state  of  the  case  then  is  this  : 

All  the  proportions  among  tracts  in  Geometry  may  be 
supplaced  by  corresponding  proportions  among  the  metric 
numbers  of  those  tracts ;  in  these  latter  proportions  we  may 
supplace  the  colon  by  the  division—,  or  quotient—,  or 
fraction-mark,  and  the  double  colon  by  the  equality-mark. 
The  ratio  of  two  tracts  we  did  not  attempt,  and  did  not 
need,  to  define  geometrically ;  but  we  now  define  the  corre- 
sponding ratio  of  the  metric  niitnbers  of  those  tracts  as  the 
quotient  of  the  one  metric  number  divided  by  the  other ; 
and  a  proportion  among  these  metric  numbers  of  tracts  we 
may  define  as  an  equality  of  ratios. 

313.  We  may  now  boldly  apply  the  ordinary  laws  of 
algebraic  equations  to  any  geometric  proportion,  understand- 
ing by  its  terms  not  the  tracts  themselves,  but  the  metric 
numbers  of  the  tracts.  The  result  will  be  some  relation 
among  the  metric  numbers  of  tracts.     If  desirable,  we  may 


Th.  CXLIII".] 


METRIC   GEOMETRY. 


237 


at  once  translate  this  relation  back  into  pure  Geometry  by 
substituting  for  the  metric  numbers  of  the  tracts  the  tracts 
themselves.  But  it  will  not  always  be  possible  to  interpret 
geometrically  the  result  of  this  substitution.  An  illustration 
will  make  this  clear. 

314.  Let/,  r,  s,  t,  u  be  the  metric  numbers  of  the  tracts 
on  which  they  are  written  in  the  A  ABC,  right-angled  at  B 
with  BD  normal  to  ^C  (Fig.  184).     Then  /=  fu  (why?), 

and         /=  /V=  (^2- j^)  (/- J-2)  ^fr'-fs- 

Whence  /V^  =/V  _^  r^^^ 

whence 


+  .-4-rV 


5^     p-"^  r" 


This  beautiful  and  important  relation  may  be  stated  thus  : 


Fig.  184. 

Theorem  CXLIII".  —  The  7-eciprocal  of  the  squared  metric 
number  of  the  altitude  to  the  hypotenuse  of  a  right  triangle 
equals  the  sum  of  the  reciprocals  of  the  squared  metric  Clum- 
bers of  the  sides. 

So  stated  the  meaning  is  intelligible  and  unmistakable. 

But  if  now  we  write  for  s,  r,p  the  tracts  themselves,  namely, 

BL)-     AB^      BC"' 

then  we  may  indeed  understand  this  relation  algebraically 
precisely  as  before,  meaning  by  the  signs  BD ,  AB ,  BC 


238 


GEOMETRY. 


[Th.  CXLIIP 


the  squared  metric  numbers  of  the  tracts  BD^  AB,  BC ; 
but  we  cannot  attach  any  geometric  meaning  to  the  equation, 
for  we  cannot  tell  what  we  mean  by  the  reciprocal  of  a  geo- 
metric square. 

315.    The   next  illustration  is  still  more  interesting  and 
important  (Fig.  185). 


Fig.  185. 

Let  ABC  be  any  A,  L  any  ray  cutting  the  sides  at  A\ 
B\  C ;  from  A,  B,  C  drop  normals  on  Z  meeting  it  at 
B,  Q,  R. 

Then  by  similar  A  we  have 

AC.BC.-.AF'.BQ, 
BA^ :  CA'  -..BQ.CR, 
CB'  :AB^  ::CB:  AP. 
If  now  we  understand  by  these  biliterals  not  the  tracts 
themselves,  but  the  metric  numbers  of  the  tracts,  the  fore- 
going proportions  will  still  hold  and  may  be  read  as  equa- 
tions and  written  thus  : 

AjC^AP    BA[^BQ     CB'  ^  CR 
BC      BQ'    CA'      CR'    AB'      AP' 


Th.  CXLIV«.]  metric   geometry.  239 

where  the  sides  of  the  equations  are  ordinary  fractions.    On 
multiplying  them  together  there  results 

BC^'CA^'AB'      '^^''^' 

Inasmuch  as  ^C  and  BC^  are  reckoned  oppositely  as 
are  also  BA^  and  CA\  BC^  and  AB\  it  is  common  and 
convenient  to  write  —  i  instead  of  i,  thus  : 

AC'-BA'-  CB'  ^  _  J 
BC'CA'-AB' 

This  is  the  celebrated  proposition  of  Menelaos  : 

Theorem  CXLIV".  —  The  continued  product  of  the  ratios 
in  which  a  ray  cut  the  sides  of  a  A  is  —  i . 

It  states  the  condition  necessary  and  sufficient  that  three 
points  on  the  sides  of  a  A  shall  be  collinear,  and  its  mean- 
ing is  perfectly  clear  so  long  as  we  mean  by  ^C,  etc.,  not 
the  tracts  but  the  metric  numbers  of  the  tracts.  But  in 
order  to  interpret  it  geometrically,  AC\  etc.,  standing  for 
the  tracts  themselves,  it  would  be  necessary  to  define  pre- 
cisely a  higher  notion,  namely,  that  of  the  volutfte  of  the 
cuboid  of  three  tracts,  and  this  would  require  us  to  pass  out 
of  our  plane  into  tri-dimensional  space. 

316.  Still  another  illustration  is  found  in  a  proposition 
the  logical  complement  of  the  preceding  (Fig.  186). 

Let  any  three  rays  through  the  vertices  of  a  A  concur  in 
O,  and  let  normals  from  O  meet  the  sides  of  the  A  AB  C  at 
the  points  A\  B\  C. 

Draw  OA,  OB,  OC,  and  form  the  pairs  of  ratios  OA' :  OB ; 
OA':OC;  OB'-.OC;  OB' :  OA  ;  OC'.OA;  OC  :  OB. 
Regarding  them  as  ratios  not  of  tracts  but  of  the  metric 
numbers  of  tracts,  we  may  treat  them  as  fractions  and  form 


240  GEOMETRY.  [Th.  CXLV. 

the  product  of  the  first  in  each  of  the  three  couplets,  and 
also  of  the   second  in  each  couplet;    these  products  are 

evidently  equal.     Now  the  fraction  ——    depends  for  its 

LJA 

value  solely  on  the  angle  a ;  hence  it  is  called  a  function  of 

G 


the  angle  a,  namely,  the  sine  of  the  angle  «,  and  so  for 
the  others.     Hence 

sine  of  a      sine  of  R      sine  of  y 
.  ■ ci  .  L  —  I  •  or 

sine  of  «'     sine  of  ^S'     sine  of  y' 

Theorem  CXLV.  —  The  cofttinued  product  of  the  ratios  of 
the  sines  of  the  angles  into  which  three  concurrent  rays 
through  the  vertices  of  a  l\  divide  the  angles  of  a  A  is  i. 

The  converse  of  this  theorem  is  easily  established,  which 
accordingly  supphes  a  test  of  the  concurrence  of  three  rays 
through  the  three  vertices  of  a  A.  Its  meaning  is  perfectly 
precise  and  unmistakable  so  long  as  not  the  tracts  but  the 
metric  numbers  of  the  tracts  are  signified  by  OA,  etc. ; 
otherwise  we  are  not  in  position  to  prove  it  nor  to  interpret 
the  symbolism  expressing  it. 


METRIC   GEOMETRY.  241 

317.  The  notion  of  sine  of  an  angle^  introduced  for  sim- 
plicity in  the  foregoing  article,  is  of  the  highest  importance 
for  all  following  geometrical  study.  But  perhaps  a  more 
fundamental  notion  is  that  of  cosine,  which  we  may  define 
thus : 

Def.  The  ratio  of  the  projection  of  a  tract  on  a  ray  to  the 
tract  itself  is  called  the  cosine  of  the  angle  between  the  ray 
and  the  tract  (Fig.  187). 


----'ar 


V 
Fig.  187. 

Thus  the  ratio  of  the  projection  /  of  the  tract  /  to  the 
tract  itself  is  the  cosine  of  the  angle  a  between  them ;  or 
/ :  /  =  a,  as  we  may  write  cosine  of  a,  which  is  commonly 
abbreviated  into  cos  a. 

318.  It  is  plain  that  the  projections  of  t  on  parallel  rays 
are  all  equal ;  hence  we  may  suppose  the  ray  of  projection 
drawn  through  the  beginning  of  /,  as  in  Fig.  188.  Then  as 
/  turns  about  o  and  «  changes  its  value,  the  projection  p  oi  t 
will  also  change.     Thus  :  for 

« =  o,  60°,  90°,   120°,   180°,  270°,  360°,  420°,  ••• 
«=  I,      h      o,     —1     —I,        o,        I,        i   ... 

319.  In  the  2d  and  3d  quadrants  the  projection  /  is 
reckoned  leftward ;  it  is  opposite  in  sense  to  the  projection 
when  a  is  in  the  ist  or  4th  quadrants,  and  accordingly  the 
cosine  is  marked  — .  When  a  increases  by  360°  (or  2  tt, 
see  Art.  336)  from  any  value,  the  revolving  tract  resumes  its 


242  GEOMETRY. 

original  position,  the  projection  resumes  its  original  value, 
and  so  too  does  the  cosine  ;  hence 

cos  (360°  +  «)=  cos  (2  7r  +  «)  =  cos«  j 

that  is,  the  cosine  is  not  changed  by  increasing  (or  decreas- 
ing) the  angle  by  a  round  angle. 

Hence,  plainly,     cos(/)t  ±  2  mi)  =  cos  a. 

Hence  the  cosine  is  called  a  periodic  function*  of  the 
angle,  the  period  being  2  tt,  that  is,  a  round  angle  (see  Art. 

336)- 

320.  If  /■  be  turned  through  any  angle  a  from  the  posi- 
tion OA,  its  projection/  is  the  same  whether  the  turning  be 
clockwise  or  counter-clockwise ;  that  is,  the  projection  is 
the  same  whether  the  angle  be  negative  or  positive ;  hence, 
too,  the  cosine  is  the  same. 

That  is,  cos  ( —  «)  =  cos  a  ; 

that  is,  the  cosine  of  the  angle  is  unchanged  by  changing  the 
sense  (or  sign)  of  the  afigle.  Now  we  learn  in  Algebra  that 
only  the  even  powers,  not  the  odd  powers,  of  a  symbol  are 
unchanged  by  changing  the  sign  of  the  symbol ;  thus  : 

(_«)2^«2^  (^-ay=a\  but  (-  aY=-a\ 

Hence  the  cosine  is  called  an  even  function  of  the  angle. 

Its  periodicity  and  its  evenness  are  the  two  cardinal  prop- 
erties of  the  cosine,  on  which  all  others  hinge. 

321.  We  may  now  arbitrarily  define  the  sine  of  an  angle 
to  be  the  cosine  of  the  complemental  angle  ;  i.e.  go°  —  az=  a  \ , 
as  we  may  write  sine  of  a,  which  is  commonly  abbreviated 
into  sin  a. 

*  Two  magnitudes  such  that  the  values  of  the  one  correspond  to  values  of 
the  other  are  C2\\&d.  functions  of  each  other. 


METRIC   GEOMETRY.  243 

Now  write  90°  —  /?  for  « ; 

we  obtain 

cos  \  90°  -  (90°  -  ^)  S  =  (90°  _  ^)  I,  or  ^  =  (90°  -  ^)  | ; 

i.e.  the  cosine  of  an  angle  is  the  sine  of  the  complemental 
angle. 

Hence  the  sine  (or  cosine)  of  either  of  two  complemen- 
tal angles,  as  «  and  y8,  is  the  cosine  (or  sine)  of  the  other ; 
i.e.  \i  a-\-  ^—  90°,  then  «  =  /?  |,  and  « |  =  ^.  When  either 
changes  by  2  tt,  so  does  the  other  oppositely ;  hence  the 
sine  as  well  as  the  cosine  returns  into  its  original  value  ;  i.e. 
the  sine  is  also  periodic  with  the  period  2  tt. 

322.  If  the  tract  /be  reversed,  that  is,  turned  through  a 
straight  angle,  its  projection  will  also  be  reversed,  but  other- 
wise unchanged  ;  hence  the  cosine  will  be  reversed  ;  that  is, 

cos(M-l-7r)  =  — cos«  j  also  cos(a  — 7r)=cos(«-f-7r)  (why?)  ; 

hence  cos  («  —  tt)  =  —  cos  «, 

or,  to  change  the  angle  by  the  half-period,  tt,  changes  the  sign 
of  the  cosine. 

323.  Since  the  cosine  is  an  even  function, 

cos  («  —  tt)  =  cos  (tt  —  «)  ; 
hence  cos  (tt  —  «)  =  —  cos  a. 

That  is,  since  a  and  tt  —  a  are  supplemental,  the  cosines  of 
supplemental  angles  are  counter — equal  in  size,  but  opposite 
in  sense  (or  sign). 

324.  Again,  if  «  -f  /?  =  90°,  then  ol\  —  ^  (why ?) . 
Then         («  +  tt) |  =  ^-tt  (why ?)  =  -  ^  =  -  « | . 

That  is,  to  change  the  angle  by  the  half-period,  tt,  changes  the 
sense  of  the  sine  as  well  as  of  the  cosine. 


244 


GEOMETRY. 

P2 


% 

^ 

-^ 

3 

/ 

\ 

V 

1 

/ 

\ 

/   ^ 

\ 

p.' 

\ 

/Pa 

R'  Po 

1 

y^o 

^^ 

i      Po' 

1/ 

K 

y 

>■ 

Fig.  i88. 


Fig.  190. 


METRIC  GEOMETRY.  245 

We  now  ask,  how  is  the  sine  affected  by  changing  the 
sense  of  the  angle  ?     We  have 

(-«)|  =  90° +«  (why?)  =  -  (90°  -  ci)  (why?)  =-(«)!; 

that  is,  the  sense  of  the  sine  changes  when  the  sense  of  the 
angle  changes.  But  this  is  the  property  only  of  odd  powers, 
not  of  even  powers,  of  a  symbol ; 

thus  (  —  ay  =  —  a^,  {—  aY  =  —a^,  etc. 

Hence  the  sine  is  called  odd  function  of  the  angle. 

Its  periodicity  and  its  oddness  are  the  cardinal  properties 
of  the  sine,  on  which  all  others  hinge. 

325.    If  q  be  the  projector  of  the  tract  /  then  -  is  plainly 

the  sine  of  the  angle  a  for  every  position  of  t.  We  may 
indeed  define  the  sine  of  a  as  equal  to  this  ratio,  and  from 
this  definition  readily  deduce  all  the  foregoing  properties 
(Figs.  188,  189,  190). 

Exercises,    i.    Prove  that  {a\y -\-  (a)^=  i. 

2.  Find  the  value  of  a  \  for  a  =  o,  30°,  45°,  60°,  90°,  1 20°, 
150°,  180°,  210°,  225°,  240°,  270°,  300°,  330°,  360°,  390°. 

3.  If  a,  b,  c,  be  the  sides  of  a  A,  a,  ji,  y  the  opposite 
angles,  ;•  the  circumradius,  prove  that 

(f'  _  b  _  c  _ 

Such  is  the  Law  of  Sines  (Fig.  191). 

4.  Prove  that  a-  =  U"-  -\-  c'  —  2  l)ca,  —  Law  of  Cosines. 

5.  If  ^  and  b  are  adjacent  sides  of  a  O,  and  ab  denote 
the  angle  between  them,  prove  that  EJ  =  ab-  ab\. 

N.B.    We  may  define  the  sine  from  this  important  theorem 
thus  :   The  sine  of  the  angle  betiveen  two  sides  of  a  CD  is  thai 


246  GEOMETRY. 

number  which  taken  as  a  multiplier  turns  the  area  of  the 
rectangle  of  the  sides  into  the  area  of  the  O. 


Fig.  191. 

326.  If  we  project  successively  the  sides  of  a  closed  poly- 
gon on  any  ray,  the  sum  of  the  projections  will  be  o  ;  for  the 
end  of  the  projection  of  the  last  side  falls  on  the  beginning 
of  the  projection  of  the  first.  This  fact  is  very  important 
in  surveying,  where  in  compassing  a  field  the  sum  of  the 
northings  must  equal  the  sum  of  the  southings,  and  these 
two  sums,  having  opposite  senses,  together  make  the  whole 
sum  o. 

So  for  the  eastings  and  westings. 

We  may  express  this  fact  in  symbols  thus  : 

j-i^i  +  j-g^a  +  >y3«3  +  •  •  •  4-  s,,c(y,  =  0=  %sa. 

Here  the  j"'s  are  the  sides,  the  a's  are  the  angles  of  the  sides 
with  a  fixed  ray,  as  the  east  and  west  line,  sa  is  the  pro- 
jection of  a  side,  and  2  is  the  symbol  of  summation. 

Exercise.  Show  by  projecting  on  a  ray  normal  to  the  first 
ray  that  ^sa  \  =  o. 

327.  If  we  project  consecutively  all  the  sides  of  a  polygon 
but  one  on  that  one,  the  sum  of  the  projections  will  be  that 
one  itself.     For  the  beginning  of  that  side  is  the  projection 


METRIC   GEOMETRY. 


247 


of  the  beginning  of  the  first,  and  its  end  is  the  projection  of 
the  end  of  the  last,  of  the  projected  sides  (Fig.  192). 


Thus,  in  a  A,  the  sum  of  the  projections  of  two  sides  on 
the  third  is  the  third. 

328.  We  make  a  most  important  appHcation  of  this  simple 
fact  in  finding  the  cosine  of  the  difference  of  two  angles 
(Fig.  193). 


Fig.  T93. 


248  GEOMETRY. 

Let  (9/*  and  OQ  make  angles  a  and  ^  with  any  fixed  ray 
OX.  Then  angle  QOF  =  a  —  p.  Now  project  any  tract 
OQ  on  OP;  there  0F=  OQ-a  —  ^.  But  instead  of  pro- 
jecting OQ  directly  we  shall  obtain  the  same  result  by  pro- 
jecting consecutively  OF  and  FQ .  The  projection  of  OF 
is  OF -a,  and  of  FQ  is  FQ-a].     But  0F=  OQ-jB,  and 

Hence     0F=:  OQ'a-(i=  OQ-a^^^  6>(2-«|-)8|. 

Or,  «-/?  =  «. /?  +  «|.y8|. 

By  changing  the  sense  of  ^  and  by  putting  90  —  a  mstead 
of  a,    let   the    student    show   that   «  +  j8=a«)8  —  a|-y8|, 

(«  +  /?)!  =  «|.^  +  .^./?l,  («-^)i=:«|.^-«r^i. 

These  four  formulae  express  the  Addition-Theorem  of  Sine 
and  Cosine. 

The  doctrine  of  Functions  of  Angles  constitutes  Trigo- 
nometry—  an  extremely  important  subject,  which  cannot 
be  pursued  any  further  here.  See  Smith's  Clew  to  Trigo- 
nometry. 

MEASUREMENT   OF   THE  CIRCLE. 

329.  Thus  far  our  linear  measurements,  or  comparisons 
of  length,  have  been  wholly  of  tracts.  The  peculiar  sim- 
plicity of  such  operations  is  due  to  the  fact  that  any  tract 
may  be  superposed  (at  least  in  thought)  on  any  other,  and 
thus  their  equality  or  inequality  infallibly  tested.  We  may 
similarly  compare  arcs  of  equal  circles,  but  not  arcs  of 
unequal  circles,  nor  arcs  and  tracts ;  for  these  cannot  be 
made  to  fit  on  each  other  to  even  the  smallest  extent.  We 
feel  sure  indeed  that  a  circle  or  arc  has  a  perfectly  definite 
length,  that  it  is  longer  than  some  tracts,  shorter  than  others, 
and  equal  to  some  others.     For  if  we  suppose  an  inextensi- 


MEASUREMENT   OF   THE    CIRCLE.  249 

ble  cord  wrapped  around  a  circular  disk,  on  unwinding  and 
straightening  the  cord  we  should  obtain  a  tract  equal  to  the 
circle  in  length.  But  it  remains  difficult  or  impossible  to 
fix  the  notion  of  the  length  or  to  determine  the  length  itself 
without  some  preliminary  definitions  and  assumptions. 

We  assume  then  that  a  circle  has  a  definite  length,  neither 
more  nor  less ;  also,  that  it  bounds  a  definite  area,  neither 
more  nor  less. 

330.  We  now  inscribe  in  the  circle  of  radius  r  a  regular 
;^-side,  and  parallel  to  this  latter  we  circumscribe  a  regular 
;?-side ;  then  bisecting  each  arc  subtended  by  a  side  of  the 
inscribed  ;^-side  we  inscribe  a  regular  2  «-side  and  also  cir- 
cumscribe parallel  to  it  a  regular  2  /z-side. 

Then  the  following  facts  are  at  once  evident : 

1 .  The  area  of  any  inscribed  polygon  is  less,  and  the  area 
of  every  circumscribed  polygon  is  greater,  than  the  area  of 
the  circle  ;  or  if  /„,  S,  and  C„  designate  these  areas,  then 

h<S<  C^  (why?). 

2.  The  area  of  the  inscribed  2  ^^-side  is  greater  than  that 
of  the  inscribed  ;/-side,  while  the  area  of  the  circumscribed 
2  ^/-side  is  less  than  that  of  the  circumscribed  ;?-side  ;  or, 

4.  >/,„  C„,<  C„  (why?). 

3.  The  area  of  each  regular  «-side  is  half  that  of  the  rect- 
angle of  the  perimeter  and  the  central  normal  on  a  side,  and 
in  case  of  the  circumscribed  polygons  this  normal  is  the 
radius  r,  but  in  case  of  the  inscribed  polygons  this  normal, 
or  apothem^  a,^,  is  less  than  r. 

4.  The  perimeters  of  inscribed  and  circumscribed  ;?-sides 
are  to  each  other  as  a^  and  r ;  for  they  are  similar  polygons, 
and  «„  corresponds  to  r. 


250  GEOMETRY. 

5.  Since  the  area  of  the  circumscribed  ;/-side  decreases 
as  n  increases,  and  since  one  dimension,  the  radius,  remains 
constant,  it  follows  that  the  other  dimension,  the  (half) 
perimeter,  must  decrease  with  increasing  n.  For  n  =  4  the 
polygon  is  a  circumsquare,  and  the  perimeter  is  8  r. 

6.  Since  the  sum  of  two  sides  of  a  A  is  greater  than  the 
third  side,  it  follows  that  the  perimeter  of  the  inscribed 
//-side  increases  with  increasing  n.  For  n  =  6  the  perime- 
ter is  6 ;-.  Hence  for  all  higher  values  of  7?.  the  perimeters 
of  both  inscribed  and  circumscribed  polygons  lie  between 
6  r  and  8  r. 

7.  Since  then  the  sum  of  the  //-sides  is  certainly  less  than 
8r,  by  making  ;z  large  enough  we  can  make  each  side, 
whether  of  inscribed  or  circumscribed  polygon,  as  small  as 
we  please,  smaller  than  one  millionth,  smaller  than  one  bil- 
lionth, smaller  than  any  assigned  magnitude  however  small. 

8.  But  the  half-side  of  the  regular  inscribed  //-side  is  a 
geometric  mean  between  the  segments  of  the  normal  diame- 
ter ;  i.e. 

S 
As  S^  is  small  at  will,  so  is  — ,  and  still  more  is  r  —  ^„, 

which  we  may  call  d^,  the  distance  between  the  parallel  sides 
of  the  inscribed  and  circumscribed  polygons. 

N.B.  A  magnitude  small  at  will  is  often  called  an  infini- 
tesimal.    Since  /'  —  a^  or  d^  is  infinitesimal  with  respect  to 

—J  which  is  itself  infinitesimal,  it  {d„)  is  called  an  infinites- 
imal of  2d  order.  But  we  are  not  now  concerned  with  this 
fact. 


MEASUREMENT   OF   THE    CIRCLE. 


251 


9.  The  difference  in  area  between  the  circumscribed  and 
the  inscribed  ;/-sides  is  a  trapezoid  the  half-sum  of  whose 
parallel  sides  is  less  than  ^r,  the  altitude  being  d^.  Since 
8  r  is  finite  and  definite  while  d^  is  small  at  will,  it  follows 
that  the  difference  in  area  of  the  circumscribed  and  inscribed 
regular  polygons  is  small  at  will. 


Fig.  194. 

10.  Again,  we  have  from  similar  homothetic  figures, 
Perimeter  of  C„  :  Perimeter  of  /„  :  \r\r—  d^. 

Hence,  dividendo, 

Perimeter  C^  :  Perimeter  C„  —  Perimeter  I,,\\r\  d^. 

But  d^  is  small  at  will ;  so  too  then  is  /?„  =  Perimeter  C„ 
—  Perimeter  of  /„ ;  i.e.  the  difference  in  perimeter  of  the 
circumscribed  and  inscribed  regular  polygons  is  small  at 
will. 

1 1 .  The  circle  lying  always  wholly  between  the  two  poly- 
gons inscribed  and  circumscribed  both  in  position  and  in 
areal  value,  it  follows  that  the  difference  between  its  area 


252  GEOMETRY. 

and  the  area  of  either  polygon  is  small  at  will ;  and  since 
the  circle  is  fixed  both  in  area  and  in  position,  it  follows 
that  the  circle  is  the  Limit  of  both  polygons,  and  that  the 
polygons  close  down  upon  it  close  at  will  as  n  grows  ever 
larger  and  larger. 

331.  Think  of  the  polygonal  strip  between  the  inscribed 
and  circumscribed  ;/-sides  as  growing  ever  narrower  and 
narrower ;  the  circle  is  a  fixed  boundary  toward  which  the 
circumscribed  ;z-side  shrinks  down  as  n  increases,  and  in 
such  a  way  that  there  is  no  assignable  point  outside  of  the 
circle,  no  matter  how  close  to  it,  that  will  not  also  fall  out- 
side of  the  circumscribed  /^-side  as  n  increases.  Likewise 
the  circle  is  the  fixed  boundary  toward  which  the  inscribed 
«-side  swells  out  as  the  ;/  increases,  and  in  such  fashion  that 
there  is  no  assignable  point  inside  of  the  circle  (no  matter 
how  close  to  it)  that  will  not  fall  inside  of  the  inscribed 
«-side  as  n  increases.  Thus  the  inscribed  and  circum- 
scribed /^-sides  close  down  upon  each  other  so  as  to  leave 
no  point  between  except  the  points  of  the  ci7'cle  itself.  As  n 
increases,  the  polygons  tend  to  absolute  coincidence  with 
each  other  and  with  the  circle.  This  fact  is  expressed  fully 
and  accurately  by  saying  that  the  circle  is  the  common 
Limit  in  length,  in  area,  in  position,  of  the  inscribed  and  cir- 
cumscribed regular  7^-sides  for  increasing  n;  it  is  expressed 
elliptically  and  inaccurately,  but  conveniently  and  frequently, 
by  saying  that  the  circle  is  or  may  be  regarded  as  a  regular 
polygon  of  a 71  infinite  number  of  sides. 

332.  The  area  of  a  circumscribed  ;z-side  is  half  the  area 
of  the  rectangle  of  the  radius  of  the  circle  and  the  perimeter 
of  the  polygon,  for  all  values  of  n.  Hence  the  area  of  the 
circle  is  half  the  rectangle  of  radius  and  the  perimeter ;  that 


MEASUREMENT  OF  ANGLES.  253 

is,  the  circle.  The  numeric  expressing  the  ratio  of  the  circle 
to  its  diameter  is  called  the  perwietric  ratio,  and  is  desig- 
nated by  the  Greek  initial  of  perimeter,  tt.  It  is  an  irra- 
tional and  hence  not  expressible  exactly  as  a  fraction, 
whether  common  or  decimal,  but  its  value  has  been  calcu- 
lated in  various  ages  to  various  degrees  of  exactitude,  and 
may  be  calculated  to  any  degree  of  exactitude.  Of  late 
years  it  has  been  calculated  (by  Shanks)  to  the  707th  decimal 
place,  and  verified  to  the  500th  —  a  degree  of  accuracy 
immensely  higher  than  can  be  attained  in  any  measurement. 
For  most  practical  purposes  the  value  tt  =  3.14159  or  even 
3. 14 1 6  is  close  enough. 

If  then  r  be  the  radius,  2  irr  is  the  length  of  the  circle, 
and  7rr  •  r,  or  ir/^,  is  its  area. 

333.  Because  the  ratio  tt  is  irrational  it  by  no  means  fol- 
lows that  it  cannot  be  constructed  geometrically ;  that  is, 
that  we  cannot  with  ruler  and  compasses  draw  a  tract  that 
shall  be  exactly  equal  to  the  circle  of  radius  r.  The  ratio 
"\/2  is  irrational,  yet  we  can  easily  construct  V2  •  r,  by 
drawing  the  diagonal  of  a  square  of  side  r.  If  we  could 
draw  a  tract  irr,  equal  to  a  half-circle,  then,  by  Problem  III^ 
p.  193,  we  could  construct  the  geometric  mean  of  r  and  nr, 
which  would  be  the  side  of  a  square  precisely  equal  to  the 
circle  in  area.  This  famous  problem  of  squiuing  the  circle 
is  therefore  not  an  irrational  one ;  it  is  unsolved,  but  possi- 
bly not  in  itself  unsolvable.   But  see  Math.  Ann.,  xx.,  p.  213. 

MEASUREMENT   OF   ANGLES. 

334.  We  have  denoted  by  tt  the  so-called  perimetric 
ratio,  namely,  of  the  circle  to  its  diameter,  2  r,  so  that,  if  r 
be  the  radius,  then  2  irr  is  the  (length  of  the)  circle,  and 
ir/^  is  its  area.     But  there  is  another  important  use  of  ir. 


254  GEOMETRY.  [Th.  CXLVI. 

335.  We  have  learned  the  ordinary  sexagesimal  division 
of  the  round  angle  into  360  equal  parts  called  degrees.  This 
artificial  umt,  degree,  does  not  recommend  itself  for  purposes 
of  mathematical  investigation,  but  a  natural  unit  is  suggested 
by  the  measurement  of  the  circle  itself.  For  it  is  plain  that 
whatever  part  an  arc  is  of  the  whole  circle,  that  same  part 
the  central  angle  of  the  arc  is  of  the  round  angle.  Thus,  if 
m  times  the  arc  make  out  the  circle,  then  m  times  the 
central  angle  will  make  out  the  round  angle ;  for,  in  add- 
ing the  arcs  about  the  centre  O,  we  at  the  same  time  add 
the  angles  at  O  subtended  by  the  arcs.  If,  however,  arc 
and  circle  be  incommensurable,  cut  the  arc  and  also  the 
angle  into  q  very  small  equal  parts.  Then  /  of  these  arc- 
parts  will  be  less  and  /  -f  i  will  be  greater  than  the  circle, 
while,  similarly,  p  of  the  angle-parts  will  be  less  and  /  +  i  will 
be  greater  than  the  round  angle ;  and  this  will  always  hold, 
no  matter  how  great  q  and  p  may  be.  Hence,  using  the 
arc  as  unit-arc  and  the  angle  as  unit-angle,  we  see  that  the 
metric  numbers  of  circle  and  round  angle  lie  always  between 

the  same  fractions,  -  and ;  and  for  increasing  /  and  q 

these  fractions  close  down  upon  each  other,  so  that  the  metric 
numbers  of  circle  and  round  angle  cannot  differ  by  ever  so 
little,  but  must  be  precisely  the  same.  If  instead  of  circle 
and  round  angle  we  take  any  other  arc  and  its  corresponding 
central  angle,  the  reasoning  remains  unchanged,  so  that  we 
have 

Theorem  CXLVI.  —  If  any  arc  and  its  corresponding 
central  angle  be  taken  as  units,  the  metric  numbers  of  any 
other  arc  (of  the  same  or  equal  circle)  and  its  corresponding 
central  angle  are  the  same. 

In  other  words, 


THE   EUCLIDIAN  DOCTRINE.  255 

Central  angles  and  their  subtending  arcs  (in  the  same  or 
equal  circle)  aie  proportional  (see  Art.  337). 

The  latter  statement  is  conciser ;  the  former  is  preciser. 

336.  Now  a  natural  unit  for  arc-measurement  is  plainly 
radius ;  hence  a  natural  unit  for  angle-measurement  is  the 
angle  whose  arc  is  radiwi^  (in  length)  ;  accordingly  we  adopt 
it  as  unit-angle  and  name  it  Radian.  The  metric  number 
of  the  circle,  radius  being  unit,  is  2  tt  ;  hence  the  metric 
number  of  the  round  angle,  radian  being  unit,  is  2  tt.  The 
radian  equals  about  57°  17'  7,7,^'. 

Corollary.  If  ;/  be  the  metric  number  of  any  angle,  and 
therefore  of  its  corresponding  arc,  radian  and  radius  being 
units,  then  of  any  other  arc,  subtending  the  same  or  equal 
angle,  but  described  with  a  radius  whose  metric  number  is 
r,  the  metric  number  will  be  nr ;  for  all  circles  are  similar. 
That  is. 

The  metric  number  of  an  arc  equals  the  product  of  the 
metric  numbers  of  its  central  angle  and  its  radius. 

Exercise.  What  are  the  natural  metric  numbers  of  a 
straight  angle  ?  A  right  angle  ?  An  angle  of  60°  ?  Of  45°  ? 
Of  30°?  Of  120°?  150°?  225°?  240°?  270°?  420°?  600°? 
720°?  1080°? 

THE    EUCLIDIAN    DOCTRINE    OF    PROPORTION. 

337.  According  to  Euclid, /<?/^r  magnitudes,  a,  b,  c,  d,  are 
in  proportion,  taken  in  order,  when  any  m-f old  of  the  first  is 
less  than,  equal  to,  or  greater  than,  any  nfold  of  the  second, 
according  as  the  same  mfold  of  the  third  is  less  than,  equal 
tOf  or  greater  than,  the  same  nfold  of  the  fourth. 


256  GEOMETRY.  [Th.  CXI«. 

In  symbols 

a\b  w  c  \  d  {?'ead  a  is  to  b  2k.s  c  is  to  d) 
when,  and  only  when, 

ifia  <  nb,  ma  =  nb,  or  ma  >  nb, 
according  as 

mc  <  7td,  mc  —  nd,  or  mc  >  nd. 

We  may  also  say  equivalently  that  a  has  the  same  ratio  to  b 
that  c  has  to  d  when,  afid  only  when,  etc. 

338.  Hereby  is  defined,  then,  precisely,  not  indeed  ratio, 
but  at  least  equality  of  ratios.  However,  it  still  remains  to 
be  proved  that  the  axiom  of  equal  magnitudes,  namely,  mag- 
nitudes equal  to  the  same  or  equal  magnitudes,  ai'e  equal  to 
eath  other,  can  be  applied  to  equal  i^atios  ;■  for  it  has  not  yet 
been  shown  that  ratios  are  magnitudes  or  may  be  treated 
as  magnitudes.  The  all-important  fact  that,  whatever  these 
ratios  may  be,  they  obey  the  axiom  of  magnitudes,  is 
expressed  in  the 

Theorem  CXI'\  —  \i  a  \  b  w  c  \  d  and  a\b\:e:f, 

then  c  :  d :  :  e  :/  (see  Theorem  CXI.). 

For  herein  is  declared  that  when  two  ratios,  c  :  d  and  e  :/, 
are  equal  to  the  same  ratio,  a  :  b,  they  are  equal  to  each 
other.  After  establishing  this  fundamental  proposition,  but 
not  before,  we  may  drop  the  double  colon,  :  :,  and  write 
a  '.  b  =  c  :  d. 

339.  We  may  illustrate  both  the  idea  and  the  method  of 
Euclid,  in  demonstrating  the  following  extremely  useful 

Theorem  CXLVII.  —  Areas  of  similar  figures  are  to  each 
other  as  the  squares  on  homologous  tracts  (in  the  figures). 


Th.  CXLVIL]       the  EUCLIDIAN  DOCTRINE. 


257 


.  Data :    F  and  F^  two  similar  figures,  /  and  /'  two  homol- 
ogous tracts,  /^  and  f^  the  squares  upon  them. 

Proof.  If  the  figures  be  curvilinear,  and  in  general  even 
if  they  be  rectihnear,  it  will  not  be  possible  to  cut  them  up 
into  corresponding  squares,  however  small,  as  is  manifest. 
Nevertheless,  we  may  cut  each  one  up  into  congruent  squares 
so  small  that  the  remainder  shall  be  less  than  any  assigned 
area,  however  small ;  that  is,  shall  be  small  at  will.  For 
(Fig.  195)  draw  two  corresponding  series  of  equidistant 
horizontals  and  verticals,  d  apart  in  i%  and  d^  apart  in  F\ 


Fig.  195. 

Let  the  extreme  verticals  in  /^  be  ^  apart,  and  let  there  be 
e;  +  I  of  them,  so  that  g—vd.  Then  the  excess  of  the  area 
of  F  over  the  sum  of  the  squares  will  be  the  sum  of  the 
pairs  of  pieces  at  the  end  of  each  vertical  strip  ;  hence  it  will 
be  less  than  2  v  little  squares,  or  less  than  a  rectangle  of  base 
2g  and  altitude  d,  i.e.  <  2gd.  But  this  rectangle  is  small 
at  will,  since  its  base  2  ^  is  constant  and  finite  while  its  alti- 


258  GEOMETRY.  [Th.  CXLVII. 

tude  d  is  small  at  will.  Similarly,  in  F\  2  ^V/'  is  small  as  we 
please. 

Now  suppose  there  are  p  little  squares  cut  out  in  F  and 
also  in  F\  where  /  is  some  very  large  but  perfectly  definite 
integer.  Then  the  areas  A  and  A'  of  F  and  F'  will  differ 
from  the  sums  of  the  squares,  pd^  and/^'^,  by  s  and  s',  two 
magnitudes  small  at  will ;  i.e. 

A=pd-'^s    and  A' =  pd''' ^  s\ 

Now  suppose  md'^  =  m^d^' ;  i.e.  the  sum  of  ni  squares  in 
F  equals  the  sum  of  ;;/  squares  in  F\     Plainly  then 

p  {md^)  =p  {m^d''')  ;    or   m  -pd^  =  m^  -pd^""  (why?). 
Now 

mA  =  m  ' pd^  +  ms  and   m^A  —  w'  -/^'^  +  m}s^ ; 
hence  mA  —  m'A'  =  ms  —  m^s\ 

But  s  and  ^'  are  small  as  we  please,  while  m  and  /«'  are 
finite  j  hence  wi-  and  m^s^  are  small  at  will ;  hence  their 
difference,  ms  —  m^s\  is  less  than  a  magnitude  small  at  will ; 
hence  it  is  o.  (Why?  Because  there  is  only  one  definite 
magnitude,  namely,  zero,  that  is  less  than  a  magnitude  small 
at  will.) 

Hence,  if  i7id^  =  wV/'^,   then   mA  =  7n^A\ 
Now  suppose  md-^7n}d^'^.   Then,  as  before,  mpd^>m'pd^^j 
and    mA  =  mpd^  -\-  ms   while    m^A'  =  m^pd^^  +  m^s\ 

Hence       mA  —  m^A^  =  {m  -pd^  —  //z'  -Z^'^)  -\-  ms  —  m's\ 
Here  again  ms  —  m's'  is  small  at  will,  while  ///  •pd'^  —  m'-pd^' 
is  some  finite  magnitude ;  hence  mA  —  mA^  is  also  some 
finite  magnitude  of  the  same  sense  ;  i.e. 

if  md^  >  m'd^^,    then    w^  >  m'A'. 
Precisely  in  like  fashion  we  prove  that 

if  md'^  <  7n'd''^j    then    mA  <  mA'. 


Th.  CXLVIP.]       the  EUCLIDIAN  DOCTRINE.  259 

That  is, 

niA  <  m^A\    mA  —  m^A',  or   fnA  <  ffi'A\ 

according  as 

md''<m'd^\    md'^  =  in'd'\    md''>7n^d'\ 
Hence,  by  definition,   the  areas  are  proportional  to  the 
squares  on  the  corresponding  tracts,  d  and  ^' ;  or 
A:  And':  d'K 
Now  compare  the  squares  on  /  and  f',  d  and  d'.     Since 
all  squares  are  similar,  and  since  d  and  d'  are  corresponding 
tracts  or  the  equals  of  corresponding  tracts  in  the  squares 
on  /  and  /',  we  have  from  the  foregoing. 

Hence,  by  the  Axiom  of  Magnitudes,  applicable  to  ratios, 

AlA'-.-./'-.r-.  Q.E.D. 

340.  Care  has  been  taken  to  conduct  the  foregoing 
demonstration  so  that  it  shall  apply  quite  as  well  to  circles, 
to  regular  triangles,  in  fact  to  any  similar  figures  drawn  on 
homologous  tracts  as  to  squares  ;  so  that  we  may  afiirm 

Theorem  CXLVII". — Areas  of  sitnilar  figures  are  pro- 
portional to  areas  of  any  similar  figures  on  homologous  tracts 
(in  the  original  similar  figures). 

341.  The  peculiar  propriety  and  advantage  of  using  the 
square  are  seen  on  stating  the  analogous  arithmetical 

Theorem  CXLVII\  —  The  metric  numbers  of  similar 
figures  are  proportional  to  the  metric  numbers  of  any  other 
similar  figures  in  the  same  ratio  of  similitude. 

Now  the  figure  (or  area)  whose  metric  number  is  easiest 
to  find  is  the  square^  whose  metric  number  is  the  second 
power  of  the  metric  number  of  its  side.     Instead  of  second 


260  GEOMETRY. 

power  it  is  usual,  almost  universal,  to  say  square  and  thus 
employ  this  latter  term  in  two  entirely  different  senses,  —  the 
proper  geometrical  sense  and  the  tropical  arithmetical  sense. 
This  double  use  of  the  term  "  square  "  is  very  regrettable  as 
being  especially  confusing  to  beginners. 

342.  We  may  now  restate  our  proposition  thus  : 

The   metric  numbers  of  two  similar  figures  are  propor- 
tional   to  the    metric  numbers  of  squares  on   homologous 
tracts ; 
Or,  are  proportional  to  the   second  powers  of  the  metric 

numbers  of  homologous  tracts  ; 
Or,  are  in  the   duplicate  ratio  of  simihtude  of  the  figures 

themselves. 
Thus,  if  the  ratio  of  similitude  be  2:5  ox  a\b,  the  ratio  of 
the  areas  will  be  4  :  25  or  ^- :  ^l  ^ 

343.  Observe  carefully  that  the  Euclidian  doctrine  of 
proportion  is  not  a  geometrical  doctrine,  but  an  arithmetical 
doctrine  applied  to  Geometry.  The  same  may  be  said  of 
the  accepted  doctrine  in  modern  texts :  it  is  Arithmetic 
applied  to  Geometry.  Nevertheless,  the  difference  between 
the  two  is  very  great.  Euclid's  is  based  wholly  on  the  oper- 
ation of  multiplication  and  employs  only  positive  integers, 
not  fractions  nor  irrationals,  which  indeed  the  Greek  did  not 
recognize  as  numbers ;  the  modern,  on  the  other  hand,  is 
based  on  the  operation  of  division,  and  necessarily  involves 
some  general  theory  of  fractions  and  irrationals.  Moreover, 
Euclid's,  while  regarded  as  cumbrous  and  very  difficult  for 
the  beginner,  is  yet  a  model  of  logical  elegance  and  rigor : 
the  Hke  can  hardly  be  said  of  the  modern  treatment.* 

*  The  usual  algebraical  treatment  of  proportion  is  not  really  sound. 

O.  Henrici,  Enc,  Brit,,  Vol.  X.,  Geometry,  §  47. 


MAXIMA   AND  MINIMA. 


261 


The  doctrine  developed  in  this  text  is  purely  geometri- 
cal, implying  no  numerical  knowledge  or  calculus.  It  is 
grounded  in  the  notions  of  parallelism  and  similarity,  to 
stand  or  fall  with  them.  Hence  it  will  be  found  to  have  no 
place  in  bi-dimensional  spherics,  the  doctrine  of  the  sphere- 
surface,  which  is  in  many  particulars  quite  analogous  to 
Planimetry,  the  doctrine  of  the  plane,  but  in  which  there  are 
no  similar  figures. 

MAXIMA   AND    MINIMA. 

344.  Already,  in  Art.  135,  the  notions  of  maximum  and 
minimum  have  been  defined,  but  it  is  well  to  add  here  that 
not  absolute  but  merely  relative  size  is  referred  to,  inasmuch 
as  a  varying  magnitude  may  pass  through  a  number  of 
maxima  and  minima,  and  of  these  some  maximum  may  be 
less   than   some   minimum.     Thus,  a   boy  may  inflate   his 


elastic  balloon  till  the  diameter  becomes  9  inches,  then  let 
it  shrink  to  a  diameter  of  7  inches,  again  inflate  it  to  a 
diameter  of  12  inches,  let  it  shrink  to  one  of  10  inches, 
again  inflate  it,  and  so  on.  Here  9  and  12  are  maxima, 
while  7  and  10  are  minima  of  the  diameter.  Plainly,  maxima 
and  minima  alternate  with  each  other,  and  the  course  of  the 


262  GEOMETRY. 

variable  may  be  depicted  by  a  waving  line,  the  values  of  the 
variable  being  the  vertical  distances  of  the  points  of  the  line 
from  a  fixed  base-line.  OF  is  the  axis  of  the  variable 
(Fig.  196);  OT  is  the  time-axis.  What  are  the  maxima 
and  minima?  How  do  the  tangents  lie  at  these  points  of 
the  curve? 

345.  The  general  doctrine  of  maxima  and  minima  calls 
for  a  method  that  shall  seize  upon  the  magnitude  in  the 
process  of  change,  and  subject  its  momentary  variations  to 
investigation.  Such  a  method  is  supphed  in  the  Infinitesimal 
Calculus.  But  there  are  many  interesting  and  important 
geometric  problems  that  yield  even  more  readily  and  com- 
pletely to  elementary  than  to  more  refined  methods,  and 
some  of  these  we  shall  now  consider. 

346.  What  is  the  maximum  parallelogram  with  given  sides  ? 
The  student  may  easily  show  it  to  be  a  rectangle. 

Corollary.  The  maximum  triangle  with  two  given  sides  is 
right-angled  between  those  sides. 

347.  What  is  the  maximum  triangle  with  given  base  and 
given  vertical  angle  ?     From  the  figure  it  is  at  once  seen  to 


Fig.  197. 


MAXIMA   AND  MINIMA.  263 

h^  symmetric  (Fig.  197).     Trace  the  variation  in  both  area 
and  perimeter  of  the  A. 

348.  What  is  the  maximum  triangle  with  given  base  and 
given  perimeter  ? 

We  are  sure  that  the  symmetric  A  is  either  maximum  or 
minimum  ;  for  as  the  vertex  sHps  either  rightward  or  leftward 
from  the  symmetric  position  by  the  same  infinitesimal  amount 
(Fig.  198),  the  two  resulting  A  are  congruent.     Hence  the 


Fig.  198. 

symmetric  A,  which  lies  between  the  two,  is  either  greater 
or  less  than  either.  That  it  is  greater,  and  hence  is  the  maxi- 
mum, is  readily  proved  thus  : 

Through  the  vertex  V  draw  a  parallel  to  the  base.  Then 
no  other  position  of  the  vertex  can  be  on  this  parallel,  as  at 
U)  for  AU^  UB  >  AV-^  VB,  as  is  seen  at  once  on  taking 
the  point  B'  symmetric  with  B  as  to  the  axis  VU.  Still  less 
can  the  vertex  take  a  position  above  the  parallel,  as  at  W. 
Hence  it  must  in  every  other  position  be  below  the  parallel, 
as  at  F' ;  2in^AVB>A  V'B  (why  ?). 


264  GEOMETRY.  [Th.  A. 

349.  Theorem  A  (Lemma).  —  If  the  base  and perwiefer 
of  a  rectili7iear  figure  be  given,  the  area  may  be  continually 
increased  by  increasing  the  number  of  sides  (besides  the  base), 
and  keeping  them  mutually  equal. 

Data :    b  the  given  base,  s  the  sum  of  the  other  sides. 

Proof.  On  b  complete  with  s  3,  symmetric  A,  a  maximum. 
On  either  side  take  a  point  D  distant  one-third  of  the  side 
AC  from  the  vertex  C.  Now  holding  the  new  base  BD 
fixed,  convert  BCD  into  a  maximum  (symmetric)  A,  keep- 
ing it  isoperitnetric,  that  is,  of  equal  perimeter.  The  result- 
ing quadrangle  ADEB  is  greater  than  the  A  ACB  (Fig. 
199),  and  has  its  three  sides  AD,  DE,  EB  equal.     Now 


drawing  AE,  we  may  proceed  similarly  with  the  A  ADE  ; 
the  resulting  5 -side  will  be  greater  than  the  4-side,  but  its 
sides  will  be  unequal,  and  we  can  still  further  enlarge  the 
figure,  keeping  it  isoperimetric,  by  equalizing  the  four  sides. 
Thus  we  may  proceed  continually,  and  at  every  step  enlarge 
the  bounded  area,  first  increasing  the  number  of  sides,  and 
then  equahzing  them.  As  long  as  any  two  consecutive 
sides  are  unequal,  we  can  join  their  ends  and  enlarge  their 
A  by  making  them  equal,     q.  e.  d. 

350.  Theorem  B  (Lemma). — Any  n-side,  the  base  being 
fixed  and  the  other  sides  mutually  equal,  has  maximum  area 
only  when  the  equal  sides  enclose  equal  angles. 


MAXIMA   AND  MINIMA.  265 

Data:  We  consider  first  a  4-side,  AB  its  base,  AC,  CD, 
DB  its  three  equal  sides,  and  the  angles  C  and  D  equal 
(Fig.  200). 


Fig.  200. 

Proof.  Suppose  the  sides  to  be  rigid  rods  hinged  at  the 
angles,  forming  a  linkage.  Precisely,  as  in  Art.  349,  we 
know  that  the  area  is  either  a  maximum  or  a  minimum,  and 
we  easily  prove  it  to  be  the  maximum  thus  : 

Deform  the  linkage  by  thrusting  the  hinge  C  down  to  O ; 
then  if  AB  >  AC,  as  C  descends  to  C  on  the  arc  of  a 
circle,  D  will  rise  to  Z>'  on  the  arc  of  an  equal  circle. 

1.  Then  t/ie  arc  CO  > arc  DD^.  For  CZ>=  CD\  and 
CZ>'  being  oblique,  its  horizontal  projection  is  <  CD', 
hence  the  lateral  or  horizontal  thrust  of  C  is  >  the  lateral 
thrust  of  D ;  that  is,  the  horizontal  projection  of  the  arc  or 
chord  CC  is  >  the  horizontal  projection  of  the  arc  or  chord 
I)D\  But  even  if  DD'  were  only  equal  to  CC,  its  hori- 
zontal projection  would  be  greater  than  that  of  CC  (why?)  ; 
hence  DD  must  be  less  than  CC. 

Corollary.    Hence   AACC>BDD'. 

2.  The  A  ICC  >  IDDK  For  if  the  fixed  length  CD  had 
slipped  with  its  ends  along  the  tangents  at  C  and  D  into  the 


266  GEOMETRY. 

•position  C"/?",  then  we  should  have  had  A/'CC"  >  VDD'^ 
(why?)  ;  much  more,  when  the  ends  shp  along  the  arcs  (or 
chords),  the  end  C  falls  lower  (to  C),  the  end  D  rises  not 
so  high  (to  Z>'),  and  the  intersection  slips  further  rightward 
(to  /),  the  large  subtractive  A  is  increased  to  ICC,  the 
small  additive  A  is  decreased  to  IDD\ 

3.  Hence  the  two  decrements  ACC  and  ICC  produced 
by  this  deformation  being  respectively  greater  than  the  two 
increments  BDD^  and  IDD\  it  follows  that  the  resultant 
area  ACD^B  is  less  than  the  original  area  ACDB.  The 
reasoning  is  not  changed  if  we  thrust  in  D  instead  of  C,  and 
it  applies  whatever  the  amount  of  the  thrust.  Hence  the 
anti-parallelogram  A  CDB  is  the  maximum. 

If,  however,  AB  <AC,  then  the  argument  about  the  arcs 
CC  and  DZ)'  is  no  longer  valid  (why?).  But  in  this  case 
it  suffices  to  use  CD  instead  of  AB  as  the  fixed  base,  and 
then  proceed  precisely  as  before. 

If  AB  =  AC,  then  the  anti-parallelogram  becomes  a  square, 
and  is  plainly  larger  than  the  rhombus  resulting  from  any 
deformation  (why?). 

Hence  in  all  cases  the  symmetric  4-side  is  greater  in 
area  than  any  asymmetric  one. 

If,  now,  in  any  ;/-side  with  {n  —  1)  equal  sides,  any  two 
consecutive  angles  between  equal  sides  be  unequal,  as  PQR 
and  QRS,  then  we  may  apply  the  preceding  reasoning  to 
the  4-side  FQRS,  and  increase  its  area  by  making  it  an 
anti-parallelogram ;  and  so  we  may  proceed  continually, 
enlarging  the  area  as  long  as  the  angles  between  the  equal 
sides  are  not  all  equal.  Moreover,  if  these  angles  be  all 
equal,  then  any  change  in  the  size  of  any  one  must  change 
the  size  of  one  adjacent,  and  hence  decrease  the  area  of  a 
4-side,  and  therewith  of  the  whole  /2-side.  Hence  the  sym- 
metric ;z-side,  with  (;/— i)  equal  sides,  (;/ —  2)  equal 
angles,  and  two  other  equal  angles,  is  the  maximum,    q.  e.  d. 


MAXIMA   AND  MINIMA.  267 

351.  Such  an  ;z-side,  however,  is  always  encyclic  (why?), 
and  we  have  seen  that  its  area  may  be  continually  increased 
by  increasing  71 ;  hence  for  no  finite  value  of  n  can  the  area 
be  an  absolute  maximum.  As  ;/  increases,  the  perimeter 
tends  accordingly  towards  a  circular  arc  as  its  limit,  and  the 
area  always  increasing  tends  towards  the  absolute  maximum 
as  its  limit.  Moreover,  the  polygonal  area  may  be  made  to 
differ  from  the  circular  by  less  than  o-  small  at  will,  while 
any  change  from  the  circular  shape  will  produce  some 
definite  decrease  in  area  at  least  equal  to  o-  (for  any  such 
change  may  be  brought  about  by  a  definite  change  however 
small  in  the  polygonal  area  followed  by  other  changes  small 
at  will) .  Hence  in  the  circular  form  the  area  attains  its 
absolute  maximum. 

352.  The  foregoing  reasoning  preserves  its  cogency  how- 
ever small  the  base  I?  may  be,  and  even  when  it  vanishes,  as 
is  manifest,  so  that  we  have  as  special  cases  : 

1.  Of  all  isoperimetric  ^/-sides  the  regular  has  maximum 
area. 

2.  A  regular  ;z-side  has  greater  area  than  the  isoperimet- 
ric regular  {n  —  i)-side. 

3.  Of  all  isoperimetric  figures  the  circle  has  maximum 
area. 

353.  The  conclusion?  reached  with'  respect  to  maximal 
areas  of  isoperimetric  figures  may  now  be  readily  converted 
into  another  set  of  conclusions  with  respect  to  minimal 
perimeters  of  equiareal  figures  ;  thus  : 

1.  Of  all  equiareal  «-sides  the  regular  has  minimum 
perimeter. 

2.  A  regular  n-side  has  less  perimeter  than  the  equiareal 
regular  ( «  —  i )  -  side . 

3.  Of  all  equiareal  figures  the  circle  has  minimal  perimeter. 


268  GEOMETRY, 

The  details  are  left  to  the  student,  who  must  remember 
that  under  fixed  conditions  increase  of  perimeter  brings 
increase  of  area  (why?). 

The  doctrine  of  Maxima  and  Minima  is  one  of  the  most 
beautiful  and  fascinating  in  the  whole  range  of  mathematics, 
and  especially  in  its  applications  in  Mechanics  it  is  of  the 
highest  practical  as  well  as  theoretical  interest. 

CONCLUDING    NOTE. 

354.  Before  closing  our  discussion  it  may  be  well  to 
recall  attention  to  certain  matters  of  vital  significance  for 
geometric  theory,  but  which  could  not  be  adequately  treated 
earlier  without  perplexing  and  even  revolting  the  student. 
The  following  sections  make  no  pretension  to  thoroughness, 
but  may  yet  enable  the  reader  to  orient  himself  properly 
in  the  subject. 

355«  Some  diversity  of  judgment  prevails  as  to  what  is 
the  simplest  form  that  can  be  given  to  the  fundamental 
assumptions  of  Geometry.  Euchd's  so-called  Axioms,  com- 
prised in  his  '  common  notions,'  Postulates,  and  Definitions, 
assume  : 

Continuity  and  the  possibility  of  Rigid  Motion  (that  is, 
of  moving  a  body  or  figure  without  changing  its  size 
or  shape)  ; 

The  Existence  of  Surfaces,  Lines,  and  Points  ; 

The  Existence  of  Planes,  Straight  Lines,  and  Circles ; 

The  Existence  of  one,  and  only  one,  Non-intersector  of  a 
straight  line  for  every  point  in  its  plane ; 

The  Equality  of  all  Right  Angles. 

This  last  may  be  proved,  and  hence  is  unnecessary.  To 
the  others  must  be  added  : 


CONCLUDING  NOTE.  269 

The  Infinity  of  the  Straight  Line,  —  not  mentioned  by 
EucUd,  but  yet  impHed  in  his  demonstrations,  and 
inserted  by  his  editors. 

Perhaps  most  moderns  would  prefer  to  assume  openly 
the  Existence  of  Plane,  Straight  Line,  and  Circle.  The 
deduction  of  these  notions  from  that  of  the  sphere,  given 
in  this  text  in  the  main  after  Bolyai  and  Frischauf,  even 
though  it  may  "  lay  no  claim  to  absolute  rigor,"  nevertheless 
seems  to  the  writer  to  be  the  most  natural  and  easy  to 
intuit. 

356.  The  term  ray  has  been  preferred  to  straight  line, 
or  'straight'  (Halsted),  because  it  seems  important  to  have 
a  single  word  for  such  a  fundamental  concept,  and  still  more 
because  the  adjective  '  straight '  involves  an  unfortunate  and 
unnecessary  assumption.  Rays  may  not  be  absolutely 
straight,  but  may  curve  with  space  itself,  and  return  upon 
themselves  like  the  Equator.  They  are  not  necessarily 
straight,  but  as  straight  as  can  be,  the  straightest  that  can 
be,  in  our  actual  space.  Think  of  one  end  of  a  short  string 
fastened  on  an  egg-shell,  and  the  string  stretched  over  the 
shell  by  a  weight  at  the  other  end.  The  string  would  then 
mark  a  straightest  (geodetic)  line  on  the  shell,  which  would 
not,  however,  be  straight. 

357.  The  infinity  of  the  ray,  though  not  openly  affirmed 
in  the  text,  is  yet  implicit  in  certain  demonstrations.  Thus, 
in  Theorems  XVIL  and  XXVIIL,  it  is  assumed  that  we 
can  lay  off  on  the  medial  ray  a  tract  equal  to  the  medial 
tract  without  returning  through  the  vertex.  But  we  can- 
not certainly  do  this  unless  the  ray  be  infinite.  A  merid- 
ian, familiar  from  Geography,  is  the  straightest  line  that 
we   can   draw   on   a   sphere,  and  corresponds  to  the  ray 


270  GEOMETRY. 

in  a  plane ;  but  if  we  go  more  than  half  a  meridian  from 
the  North  pole,  and  then  go  on  as  far-  again  on  the  same 
meridian,  we  shall  always  return  through  the  same  North  pole. 
Hence  the  demonstrations  in  the  text  fail  in  generahty,  if 
the  ray  be  of  finite  length.  Hence,  too,  the  familiar  and 
plausible  Theorem  XXHI.,  B,  that  from  a  point  without  a  ray 
only  one  normal  can  be  drawn  to  the  ray,  also  fails  unless  the 
ray  be  infinite  ;  for  the  proof  of  it  rests  on  Theorem  XVH. 
In  fact,  on  a  sphere  all  meridians  from  a  certain  point, 
the  pole  (North  or  South),  are  normal  to  the  Equator,  the 
straightest  line  on  the  sphere.  In  like  manner  the  reasoning 
of  Arts.  *65  and  *66,  after  Bolyai  and  Frischauf,  is  seen  to 
assume  the  infinity  of  the  ray  and  the  plane,  —  let  the  stu- 
dent show  at  what  points. 

358.  But  in  the  more  general  disctdssion  of  Arts.  67-71 
all  such  assumptions  —  as  well  as  Axiom  7,  that  two  rays 
can  meet  in  only  one  point,  which  is  kjiown  to  hold  only 
for  the  comparatively  small  region  of  our  experience  —  are 
dropped,  and  the  student  must  note  with  the  utmost  care 
that  four  possible  space-forms  result  from  our  refusal  to 
make  these  assumptions  : 

A.  If  the  rays  LM  and  VM\  isoclinal  to  the  third  ray  or 

transversal  T,  meet  in  two  points,  on  the  right  and  on 
the  left,  then  we  have  so-called  double  Elliptic  space. 

B.  If  they   meet  in  one  point  only,  then  we  have  simple 

Elliptic  space. 

C.  If  they  do  not  meet  at  all,  but  if  there  be  also  other 

non-intersectors  through  the  point  6>'  (that  is,  if  we 
grant  Axiom  A  but  reject  Axiom  B),  then  we  have 
Hyperbolic  space. 

D.  Lastly,   if  we  grant  both  Axiom  A  and  Axiom  B,  then 

we  have  ordinary  Parabolic  space. 


CONCLUDING  NOTE.  11\ 

359.  The  names  Elliptic,  Hyperbolic,  Parabolic  have 
been  given  by  Klein.  They  mean  lacking^  exceeding,  equal- 
ling, and  refer  to  a  certain  characteristic  magnitude  called 
the  Riemannian  '  measure  of  curvature '  (Riemann'sche 
Kruemmungsmaass) ,  which  in  the  three  cases  is  respec- 
tively negative,  positive,  o,  or  less  than  o,  greater  than  o, 
equal  to  o.  Instead  of  Klein's  terms  we  sometimes  meet 
with  Riemannian,  Lobatschevskian  (or  Gaussian),  and 
Euclidian,  from  Riemann,  Lobatschevsky  and  Gauss,  and 
Euclid,  —  mathematicians  that  first  set  forth  clearly  the 
properties  of  the  space-forms. 

360.  Some  of  the  distinguishing  features  of  these  four 
spaces  are  the  following  : 

A.  I.  The  ray  is  closed  and  finite. 

2.  The  sum  of  the  angles  in  a  plane  A  is  >  a  straight 

angle. 

3.  Two  rays  that  meet  in  one  point  meet  also  in  a  sec- 

ond point. 

B.  I.  The  ray  is  closed  and  finite. 

2.  The  sum  of  the  angles  in  a  plane  A  is  >  a  straight 

angle. 

3.  Two  rays  meet  at  most  in  one  point  only. 

C.  I.  The  ray  is  not  closed,  but  infinite. 

2.  The  sum  of  the  angles  in  a  plane  A  is  <  a  straight 

angle. 

3.  Two  rays  meet  at  most  in  one  point  only. 

D.  I.  The  ray  is  not  closed,  but  infinite. 

2.  The  sum  of  the  angles  in  a  plane  A  is  =  a  straight 

angle. 

3.  Two  rays  meet  at  most  in  one  point. 


272  GEOMETRY. 

361.  It  is  curious  and  noteworthy  that  the  ray  in  a  sim- 
ple Riemannian  plane  cuts  the  plane  through,  but  not  in 
two.  For,  take  any  ray  R  and  two  points  close  together,  P 
on  the  left  and  Q  on  the  right  of  R ;  through  P  and  Q 
draw  a  ray  meeting  R  at  M.  Then  we  may  pass  from  P  to 
Q  rightward  through  M\  or,  since  the  rays  are  closed  and 
meet  only  in  M,  we  may  pass  from  P  \.o  Q  leftward  and 
not  through  M\  i.e.  we  may  pass  from  one  side  of  the  ray 
R  to  the  other  without  crossing  it.  This  may  be  hard  for 
us  to  imagine,  but  perhaps  not  harder  than  for  the  ancients 
to  imagine  antipodes.  Think  of  a  hollow  ring  —  a  circle 
running  all  round  it  or  across  it  would  cut  it  through,  but 
not  in  two.  The  Riemannian  plane  is  not  such  a  ring,  but 
is  ring-hke  in  being  thus  doubly  compendent  (Art.  162). 

362.  It  will  be  well  for  the  student  to  observe  carefully 
just  where  the  proof  of  Theorem  XXXI.  breaks  down  on 
rejecting  Axiom  B.  We  may  then  still  draw  through  C  a 
non-intersector  of  AB,  making  the  alternates  a  and  a'  equal ; 
and  we  may  also  draw  through  C  a  non-intersector  of  AB, 
making  the  alternates  fi  and  ^'  equal.  But  in  the  absence 
of  Axiom  B,  we  cannot  know  that  these  two  non-inter- 
sectors  are  the  same ;  hence  we  cannot  know  that  the  sum 
a'  +  7  +  )8'  =  a  straight  angle  ;  hence  we  cannot  affirm 
that  the  sum  a  -f  y  +  i^  =  a  straight  angle.  In  fact,  if  the 
two  non-intersectors  are  not  the  same,  then  manifestly 
'-i  +  7  +  i3  =  «'  +  y  +  ^'  <  a  straight  angle. 

363.  Actual  measurement,  direct  and  indirect,  of  the 
angles  of  a  A  yields  a  sum  always  very  near  to  a  straight 
angle.  But  even  the  largest  A  we  can  construct  and  com- 
pute in  the  heavens  are  yet  extremely  small  relatively  to  the 
whole  of  space,  whether  space  be  finite  or  infinite  ;  and  since 
the  defect  under  a  straight  angle,  or  the  excess  over  a  straight 


CONCLUDING  NOTE.  273 

angle,  if  there  be  any  such  defect  or  excess,  must  vary  with 
the  size  of  the  A,  in  observed  A  it  would  be  extremely  small 
and  so  might  elude  our  observation.  In  fact,  for  extremely 
small  A,  the  only  A  of  experience,  the  four  spaces  are  so 
nearly  alike  in  properties  as  to  be  indistinguishable ;  just  as 
if  the  earth's  radius  were  a  decillion  times  as  great  as  it  is, 
and  our  experience  extended  over  no  more  than  its  present 
surface,  we  should  be  unable  to  say  whether  it  was  flat,  or 
sphere-shaped,  or  egg-shaped,  or  ring-shaped,  or  saddle- 
shaped. 

364.  It  appears  then  that  the  natural  question.  Which  of 
the  four  possible  homoeoidal  spaces  is  our  actual  space  ?  is 
at  present  unanswerable.  Our  experience  is  still  too  narrow 
to  enable  us  to  decide  or  even  to  conjecture.  Why,  then, 
do  we  seem  to  prefer  parabolic  space,  and  build  up  our 
geometries  on  Euclid's  foundations  ?  Because  it  is  easier, 
more  convenient.  The  superior  simplicity  of  the  Euclidian 
geometry  is  conspicuous  in  its  doctrine  of  the  parallel,  the 
unique  intersector,  and  of  the  sum  of  the  angles  in  the  plane 
A,  which  is  a  constant,  the  straight  angle.  The  ground  of 
our  preference,  then,  is  not  a  logical,  but  an  economical  one. 

365.  Lastly,  let  the  student  never  forget  that  the  question 
as  to  the  fundamental  properties  of  our  space  is  at  bottom  a 
question  as  to  the  constitution  of  our  own  minds.  It  is  they 
that  at  every  instant  project  images  of  their  own  states  and 
of  all  beings  as  related  to  them,  and  build  up  these  pro- 
jections into  the  world  of  phenomena  about  us,  which  we 
call  space  and  its  contents.  Space,  then,  is  made  the  way 
our  spirits  make  it,  and  to  know  its  fundamental  properties 
is  to  know  fundamentally  the  mode  in  which  the  spirit 
objectifies  to  itself,  makes  an  object  of  its  own  contempla- 
tion, the  world  of  Not-self  about  it.    Whence  it  appears  that 


274  GEOMETRY. 

not   only   all   physical   problems,  but  also  all  geometrical 
problems,  root  finally  in  Metaphysics. 

366.  In  the  writer's  judgment  the  doctrine  of  non- 
euclidian  spaces  and  of  hyper-spaces  in  general  possesses 
the  highest  intellectual  interest,  and  it  requires  a  far-sighted 
man  to  foretell  that  it  can  never  have  any  practical  importance. 

The  student  who  would  pursue  the  subject  should  read 
Halsted's  excellent  translations  of  Lobatschevsky  and  Bolyai, 
the  Lectures  and  Addresses  of  Clifford  and  Helmholtz, 
Ball's  article  on  Measurement  in  the  Encyclopaedia  Britan- 
nica,  and  afterwards  the  monographs  of  Riemann,  Klein, 
Newcomb,  Beltrami,  Killing,  and  should  also  consult  the 
bibliography  of  the  subject  as  given  by  Halsted  in  the 
American  Journal  of  Mathematics y  Vols.  I.  and  II. 

EXERCISES   V. 

1 .  Central  rays  of  a  parallelogram  bisect  it,  and  conversely. 

2.  Central  rays  of  any  closed  centrally  symmetric  figure 
bisect  it,  and  conversely. 

3.  Tracts  bisecting  the  mid-parallel  of  a  trapezoid  and 
ending  in  its  parallel  sides  bisect  it,  and  conversely. 

4.  The  sums  of  the  opposite  A,  into  which  tracts  from  a 
point  within  a  parallelogram  to  its  vertices  divide  it,  are 
equal. 

5.  The  sums  of  the  opposite  A  formed  by  tracts  from 
any  point  of  the  mid-parallel  to  the  vertices  of  a  trapezoid 
are  equal. 

6.  Tracts  from  a  vertex  of  a  regular  6-side  to  the  other 
vertices  and  the  mid-points  of  the  remotest  sides  divide  it 
into  six  equal  A. 

7.  The  A  whose  vertices  are  the  ends  of  one  obHque 


EXERCISES    V.  275 

side  of  a  trapezoid  and  the  mid-point  of  the  other  is  half 
the  trapezoid. 

8.  The  A  of  the  medials  of  a  A  has  f  of  the  area  of  the  A. 

9.  /'is  a  point,  ABCD  a  parallelogram  ;  then  AAPC  = 
A  AFD  ±  A  AFB. 

10.  The  joins  of  the  mid-points  of  the  opposite  sides  of  a 
4-side  concur  with  the  join  of  the  mid-points  of  the  diagonals. 

11.  *  Divide  a  A  or  parallelogram  into  two  parts  in  the 
ratio  /:;//. 

12.  Divide  a  parallelogram  by  tracts  from  a  vertex  into 
n  equal  parts. 

13.  Divide  a  A  into  n  equal  parts  by  tracts  from  a  point 
on  a  side. 

14.  Transform  a  A  or  parallelogram  into  another  of  same 
base  with  a  given  angle  at  the  base. 

15.  Transform  a  A  or  parallelogram  into  another  with 
given  base. 

16.  Transform  a  A  or  parallelogram  into  another  with 
given  base  and  given  adjacent  angle. 

17.  Transform  a  trapezoid  into  an  anti-parallelogram  of 
the  same  altitude. 

18.  The  common  border  of  two  areas  lying  between  two 
rays  is  a  broken  line ;  rectify  this  border  without  affecting 
the  areas. 

19.  Inscribe  in  a  given  circle  a  rectangle  of  given  area. 

20.  If  tracts  between  the  parallel  sides  of  a  trapezoid,  not 
intersecting  within  it,  divide  the  mid-parallel  into  n  equal 
parts,  they  also  divide  the  area  into  n  equal  parts. 

21.  In  a  trapezoid,  the  join  of  the  mid-points  of  the 
diagonals  equals  the  difference  of  the  parallel  sides. 

22.  Construct  two  tracts,  knowing  their  ratio  and  their 
sum  or  difference. 

23.  Investigate   the   proportionaHties   between   the   seg- 


276  GEOMETRY. 

ments  of  the  altitudes  of  a  A  made  by  the  orthocentre  and 
the  segments  of  the  sides  made  by  the  altitudes. 

24.  To  the  base  of  a  A  draw  a  parallel  cutting  off  a  A  of 
given  perimeter. 

25.  *  The  sum  of  two  similar  figures  on  the  sides  of  a  right 
A  equals  a  third  similar  figure  on  the  hypotenuse. 

26.  If  intersecting  semicircles  be  drawn  on  all  the  sides 
of  a  right  A,  the  sum  of  the  two  small  crescents  will  equal 
the  larger  (Hippocrates,  450  b.  c). 

27.  Bisect  a  A  by  a  parallel  to  its  base,  and  generalize 
the  problem. 

28.  The  rectangle  of  the  segments  into  which  one  alti- 
tude of  a  A  is  cut  by  the  others  is  the  same  for  all  the 
altitudes. 

29.  Construct  a  A,  knowing  its  altitudes. 

30.  The  medials  of  a  A  cut  it  into  six  equal  A. 

31.  A's  mid-rays  cut  sides  into  irjslt ;  1tJ=iJ-\-jl-\-  li=  ^rs. 

32.  The  altitudes  of  a  A  cut  it  into  six  A,  so  that  the 
sums  of  the  alternate  A  are  equal. 

33.  Through  the  perimeters  of  an  inscribed  and  a  circum- 
scribed regular  /z-side  express  those  of  the  2  ;z-side. 

34.  The  area  of  a  ring  between  two  concentric  circles 
equals  that  of  a  circle  having  as  diameter  the  chord  of  the 
outer  circle  tangent  to  the  inner  circle. 

35.  The  radius  of  the  earth  being  6370  kilometers,  how 
far  might  one  descry  a  ship  from  the  summit  of  Chimborazo 
(21,424  feet)  ? 

36.  How  many  hills  of  corn  one  yard  apart  may  be 
planted  in  a  rectilinear  field  of  one  acre  ? 

37.  Chords  from  any  point  of  a  circle  to  the  ends  of  a 
diameter  divide  its  conjugate  chords  harmonically. 

38.  Chords  from  any  point  of  a  circle  to  the  ends  of  a 
chord  divide  its  conjugate  diameter  harmonically. 


EXERCISES    V.  277 

39.  Transform  a  A  into  another  A'  similar  to  a  given  A". 

40.  Find  the  locus  of  a  point  whose  distances  from  two 
given  rays  are  in  a  fixed  ratio. 

41.  All  rays  whose  distances  from  two  fixed  points  are  in 
a  fixed  ratio  envelop  two  fixed  points  ;  find  them. 

42.  On  a  given  ray  find  a  point  whose  distances  from 
two  fixed  points  (or  rays)  are  in  a  fixed  ratio. 

43.  Through  a  given  point  draw  a  ray  whose  distances 
from  two  fixed  points  shall  be  in  a  fixed  ratio. 

44.  Find  a  point  (or  points)  whose  distances  from  three 
fixed  non-concurrent  rays  L,  M,  iV shall  be  as  /:  w  :  71. 

45.  Find  a  ray  (or  rays)  whose  distances  from  three  non- 
collinear  points  A,  B,  C  shall  he  2iS  a-,  b  -.c. 

46.  Find  a  point  whose  distances  from  three  fixed  points 
shall  he  2iS  I :  m  '.  n. 

47.  In  every  trapezoid  the  mid-points  of  the  bases  along 
with  the  intersections  of  the  non-parallel  sides  and  of  the 
diagonals  form  an  harmonic  range. 

48.  The  locus  of  a  point  whose  distances  from  A  and  B 
are  in  the  ratio  ^  :  ^  is  a  circle  on  AB  as  central  ray  and 
dividing  AB  harmonically  in  the  ratio  a  :  b. 

49.  The  locus  of  a  point  whence  the  collinear  tracts  AB 
and  BC  appear  equal  is  a  circle  dividing  AC  harmonically. 

50.  Find  the  point  whence  three  collinear  tracts  AB, 
BC,  and  CD  appear  equal. 

51.  /'is  a  given  point  in  a  given  angle  BAC \  draw  BC 
so  ihdXPB.BC'.  :l\m,  or  BP.PCwl.m,  or  BA.AC:: 
l\  m. 

52.  From  the  continued  ratio  of  the  sides  determine  the 
continued  ratio  of  the  altitudes  of  a  A. 

53.  A  marble  rests  in  a  conical  glass  ;  another  rests  on  it 
and  touches  the  glass  all  around ;  and  another,  and  so  on. 
How  are  the  radii  of  the  marbles  related  in  size  ? 


278  GEOMETRY. 

54.  Find  the  area  bounded  by  three  equal  tangent  circles. 

55.  Compare  the  A  of  the  centres  with  the  A  of  the  com- 
mon tangents  of  three  tangent  circles. 

56.  In  a  regular  A  is  inscribed  a  circle ;  tangent  to  it 
and  two  sides  of  the  A,  another  circle  ;  and  so  on.  How  do 
the  radii  decrease? 

57.  J/  is  the  mid-point  and  P  any  other  point  of  the 
tract  AB ;  semicircles  are  drawn  on  AM,  MP,  AP,  and  PB, 
all  on  the  same  side  of  AB ;  show  that  the  sum  of  the  first 
and  second  equals  the  fourth  plus  the  area  bounded  by  the 
first  three. 

58.  Inscribe  a  circle  in  a  given  sector  of  a  circle. 

59.  Find  a  point  on  a  circle  whose  distances  from  two 
chords  are  proportional  to  the  chords. 

60.  The  distance  of  any  point  of  a  circle  from  the  chord 
of  contact  of  two  tangents  is  a  mean  proportional  between 
its  distances  from  the  tangents. 

61.  When  does  the  altitude  to  the  hypotenuse  of  a  right 
A  divide  the  hypotenuse  in  extreme  and  mean  ratio  ? 

62.  In  a  regular  5-side  each  diagonal  is  divided  in  extreme 
and  mean  ratio  by  two  others  and  all  form  a  regular  5-side. 
Compare  the  areas  of  the  two  5 -sides. 

63.  The  rectangle  of  the  distances  of  any  point  of  a  circle 
from  two  opposite  sides  of  an  encyclic  4-side  equals  the 
rectangle  of  the  distances  from  the  diagonals. 

64.  RA  and  RB  are  two  equal  rigid  rods  pivoted  at  R ; 
6>(2  is  a  rigid  rod  pivoted  2ii  O  dJid  0Q=  OR ;  AQBP  is 
a  rhombus  formed  of  equal  rigid  rods  movable  about  their 
junction-points.  Show  that,  as  Q  traces  a  circle  about  O, 
P  traces  a  ray  normal  to  the  ray  OR,  and  that  P  and  Q  are 
inverse  as  to  R. 

N.B.  Such  is  the  mechanical  invertor  called  Peaucellier's 
ceU. 


EXERCISES    V.  279 

65.  In  any  range  of  four  points,  as  ABCPj  show  that 
AB'  CF+BC'AF^  CA-  BP=o. 

66.  In  any  pencil  of  four  tracts,  as  OA,  OB,  OC,  OD, 
show  that 
AAOB-ACOD-\-ABOC'AAOD-^ACOA'ABOD=o. 

67.  The  diagonals  of  a  parallelogram  concur  with  the 
diagonals  of  its  complemental  parallelograms. 

68.  The  mid-points  of  the  diagonals  of  a  4-side  are 
collinear. 

69.  Rays  through  the  vertices  of  a  A  and  the  points  of 
touch  of  the  ex-circles  concur. 

70.  Normals  to  the  sides  of  a  A,  at  the  points  of  touch  of 
the  ex-circles,  concur. 

71.  When  normals  from  the  vertices  of  one  A  to  the 
sides  of  another  A'  concur,  so  do  normals  from  the  vertices 
of  A'  to  the  sides  of  A. 

72.  Normals  to  the  three  sides  of  a  A  through  the  points 
of  touch  of  two  ex-circles  and  the  in-circle  concur. 

73.  The  feet  of  normals  from  any  point  of  a  circle  to  the 
sides  of  an  inscribed  A  are  collinear  (on  Simson's  Line). 

74.  Rays  through  the  vertices  of  a  A  and  the  points  of 
touch  of  the  in-circle  concur. 

75.  Mid-rays  of  the  angles  of  a  A  (three  outer,  or  two 
inner  and  one  outer)  intersect  the  opposite  sides  colHnearly. 

76.  Tangents  to  the  circumcircle  of  a  A  at  its  vertices 
intersect  the  opposite  sides  collinearly. 

77.  If  the  joins  of  the  vertices  of  a  A  with  three  inter- 
sections of  the  opposite  sides  by  a  circle  concur,  so  do  the 
joins  of  the  vertices  with  the  other  three  intersections  of 
the  opposite  sides. 

78.  Find  a  circle  as  to  which  two  pairs  of  collinear  points 
are  inverse.  {Hi?tt  Draw  a  circle  through  each  pair,  and 
also  their  power-axis.)     When  is  the  circle  sought  real? 


280  GEOMETRY. 

79.  Find  the  inverse  of  a  given  point  as  to  a  given  circle. 

80.  Given  two  points,  find  a  circle  of  inversion  having 
given  radius  or  given  centre. 

81.  A  circle  through  a  pair  of  inverse  points  cuts  the 
circle  of  inversion  orthogonally,  and  conversely. 

82.  The  squared  distances  of  a  point  on  the  circle  of 
inversion  from  two  inverse  points  are  in  the  ratio  of  the 
central  distances  of  the  points. 

Z-^.  The  power-axis  of  a  fixed  circle  and  a  variable  circle 
through  two  inverse  points  as  to  the  fixed  circle  envelops 
a  fixed  point. 

84.  Find  the  inverse  of  a  circle  when  it  passes  through 
and  also  when  it  does  not  pass  through  the  centre  of 
inversion. 

85.  When  does  a  circle  invert  into  itself  ? 

86.  A  circle,  its  inverse,  and  the  circle  of  inversion  have 
a  common  power-axis. 

87.  Inversion  does  not  change  the  size  of  the  angle  under 
which  two  lines  intersect. 

ZZ.  The  9-point  circle  of  a  A  touches  the  in-  and  ex-circles 
of  the  A. 

89.  The  joins  of  the  intersections  of  two  circles  with  their 
common  diameter  and  a  common  orthogonal  circle  concur 
on  that  orthogonal. 

90.  The  join  of  the  polars  of  two  points  is  the  pole  of  the 
join  of  the  points. 

91.  If  a  pole  trace  the  sides  of  an  ;7-side,  P,  the  polar 
will  envelop  the  vertices  of  an  //-angle,  Q\  and  if  the 
pole  trace  the  sides  of  Q,  the  polar  will  envelop  the  vertices 
oiP. 

92.  If  the  pole  trace  any  line  Z,  the  polar  will  envelop 
a  corresponding  line  V ;  and  if  the  pole  trace  Z',  the  polar 
will  envelop  Z. 


EXERCISES    V.  281 

Defs.  Two  lines,  either  of  which  is  enveloped  by  the 
polar,  as  the  other  is  traced  by  the  pole,  are  called  reciprocal. 

When  the  reciprocals  coincide,  the  figure  is  called  self- 
reciprocal  or  self-conjugate,  while  the  centre  and  the  circle 
of  reference  are  called  the  polar  centre  and  the  polar  circle 
of  the  figure. 

93.  If  a  A  has  a  polar  centre,  it  is  the  orthocentre. 

94.  If  the  polar  circle  is  real,  the  A  is  obtuse-angled. 

95.  The  polar  circle  of  a  A  is  orthogonal  to  the  circles 
on  the  sides  as  diameters. 

96.  Invert  the  sides  of  a  A  as  to  its  polar  circle. 

97.  The  ends  of  a  diameter  of  a  circle  are  conjugate  as 
to  every  orthogonal  circle. 

98.  As  the  orthogonal  circle  (of  97)  varies,  the  polar  of 
either  diameter-end  envelops  the  other. 

99.  The  distances  of  any  two  points  from  a  polar  centre 
vary  as  the  distances  of  each  from  the  polar  of  the  other  as 
to  that  centre  (Salmon) . 

100.  Polar  reciprocal  A  are  in  perspective. 

Def.  Circles,  every  pair  of  which  have  the  same  power- 
axis,  are  called  co-axal. 

loi.  If  two  circles  intersect  in  A  and  B,  all  co-axals  go 
through  A  and  B. 

102.  The  contrapositive  of  loi. 

Hence  there  are  two  kinds  of  co-axals  :  common  point 
co-axals  (loi)  and  non-intersectors,  so-called  limiting  point 
co-axals  (102). 

103.  The  centre  line  of  common  point  co-axals  is  a  ray, 
namely,  the  common  power-axis  of  co-axals  orthogonal  to 
the  common  point  co-axals. 

104.  One  and  only  one  of  these  orthogonals  goes  through 
every  point  of  the  plane  except  the  common  points,  which  are 
therefore  called  (Poncelet)  limiting  points  of  the  orthogonal 
co-axals. 


282  GEOMETRY. 

105.  Draw  a  double  system  of  mutually  orthogonal  co- 
axals. 

106.  The  polars  of  a  point  as  to  co-axals  concur,  and 
conversely. 

107.  The  difference  of  the  powers  of  a  point  as  to  two 
circles  is  proportional  to  the  distance  of  the  point  from  the 
power-axis. 

108.  The  locus  of  a  point  whose  tangent  lengths  from  two 
circles  are  in  fixed  ratio  is  a  co-axal  circle. 

109.  When  does  the  power-centre  of  three  circles  become 
indefinite,  and  when  does  the  radius  become  imaginary? 

1 10.  The  polar  centre  of  a  A  is  the  power-centre  of  three 
circles  on  any  tracts  from  the  three  vertices  to  the  sides  as 
diameters. 

111.  Three  collinear  points  on  the  sides  of  a  A  being 
joined  with  the  vertices,  the  circles  on  these  joins  as  diameters 
are  co-axal. 

112.  The  four  polar  centres  of  the  four  A,  formed  by  the 
sides  of  a  4-side  taken  in  threes,  are  collinear  on  the  power- 
axis  of  circles  on  the  diagonals  of  the  4-side  as  diameters. 

113.  Invert  a  double  orthogonal  system  of  co-axals;  as 
special  case  take  a  common  or  Umiting  point  as  centre. 

114.  Show  that  any  two  circles  may  be  inverted  into  equal 
circles,  and  find  the  locus  of  the  centre  of  inversion. 

115.  Invert  three  non-co-axal  circles  into  three  equal 
circles. 

116.  Draw  a  circle  touching  these  three  equal  circles,  and 
re-invert  the  four  circles.     What  problem  is  hereby  solved  ? 

117.  Discuss  the  various  possible  positions  of  the  centres 
of  similitude  of  two  circles. 

Def.  The  circle  of  similitude  of  two  circles  has  the  tract 
between  the  centres  of  similitude  as  diameter. 

118.  The  circle  of  similitude  is  co-axal  with  the  two  circles. 


EXERCISES   V.  283 

119.  From  any  point  on  the  circle  of  similitude  the  two 
circles  appear  to  be  of  the  same  size. 

120.  Find  a  circle  as  to  which  the  power-axis  and  the 
circle  of  simiUtude  invert  into  each  other. 

Defs.  A  tract  with  its  ends  on  a  figure  is  called  a  chord 
of  the  figure.  A  ray  bisecting  a  system  of  parallel  chords  is 
called  a  diameter  of  the  figure.  Two  diameters,  each  halv- 
ing all  chords  parallel  to  the  other,  are  called  conjugate. 

121.  Every  two-ray  (^Zweistrahl),  or  angle  considered  not 
as  a  magnitude  but  as  a  figure,  has  an  infinity  of  diameters, 
namely,  every  ray  through  its  vertex. 

122.  A  A  has  three  diameters,  namely,  its  medials. 

123.  A  parallelogram  has  two  pairs  of  conjugate  diameters. 

124.  A  two-ray  has  an  infinity  of  conjugate  diameters. 

125.  If  Z' and  ^' be  conjugate  diameters  of  (the  two- 
ray)  LM,  then  L  and  M  are  conjugate  diameters  of  (the 
two-ray)  L'M'. 

Def.   Four  such  rays  are  called  harmonic,  because  : 

126.  They  cut  every  transversal  in  four  harmonic  points. 

127.  Conversely,  four  concurrent  rays  through  four  col- 
linear  harmonic  points  are  harmonic. 

128.  The  join  of  an  outer  vertex  with  the  intersection  of 
the  inner  diagonals  of  a  4-side  cuts  two  opposite  sides,  each 
in  a  fourth  harmonic  to  the  three  vertices  on  the  side. 

Hint.  In  Fig.  54,  let  /  be  the  intersection  of  the  inner 
diagonals  CE,  DF;  let  a  4th  harmonic  through  A  cn\.  BC 
and  BD  at  H  and  K.  Then  IV,  IE,  IB,  IK  are  four  har- 
monic rays,  and  so  are  IF,  IC,  IB,  IB;  also  ID,  IE,  IB, 
are  the  same  rays  as  IF,  IC,  IB  ;  hence  IH  and  IK  are  the 
same  ray ;  hence  the  4th  harmonic  through  A  goes  through  /. 

129.  Enumerate  the  harmonic  ranges  and  pencils  in  128. 

130.  A  system  of  co-axals  determines  the  points  of  every 
ray  in  pairs  of  conjugates,  F  and  F',  Q  and   Q,  so   that 


284  GEOMETRY. 

IP  •  IP'  =  IQ  •  IQ',  where  /  is  the  intersection  of  the  ray 
with  the  power-axis. 

Defs.  Points  so  determined  are  said  to  form  an  Involution. 
The  fixed  point  /  is  called  the  centre  of  the  Involution,  the 
constant  product  or  rectangle  of  the  central  distance  of  the 
conjugates  is  called  the  power  of  the  Involution,  and  is  posi- 
tive or  negative  according  as  the  distances  are  like-sensed 
or  unlike-sensed. 

131.  An  Involution  is  determined  by  its  centre  and  a 
pair  of  conjugates,  or  by  two  pairs  of  conjugates. 

132.  When  the  power  is  positive  there  are  two  self-conju- 
gate points  (called  double  points  or  foci),  and  the  focal  tract 
is  divided  harmonically  by  every  pair  of  conjugates. 

133.  Conversely y  all  pairs  of  points  dividing  a  tract  FF^ 
harmonically  form  an  Involution  with  F  and  F  as  foci  and 
the  mid-point  /  of  FF^  as  centre. 

134.  When  the  power  is  negative  there  are  no  (real)  foci 
but  there  are  two  conjugate  points,  E  and  E\  equidistant 
from  the  centre. 

Def.  Rays  of  a  pencil  passing  through  an  Involution  of 
points  form  an  Involution  of  Rays. 

135.  Develop  and  express  the  reciprocal  properties  cor- 
responding to  1 3 1-4. 

136.  In  ge?ieral,  the  points  of  a  row  and  the  intersections 
of  their  polars  with  the  axis  of  the  row  form  an  Involution. 

Hint.  P  the  point,  L  the  axis,  O  the  centre  of  the 
referee  circle  S,  which  cuts  L  at  i^and  F\  Q  the  intersection 
of  L  with  the  polar  of  P,  01=  d=  distance  of  L  from  O, 
OA  =  r=  radius  of  6*  on  OF.  Draw  a  circle  K  on  FQ  as 
diameter  about  C  cutting  OF  at  R.  Then,  difference  of 
powers  of  O  and  /  as  to  A"  is 

OC'  -JC'=~0~f  =  r'-IQ'IF. 

Hence   r-  —  d'^  =  IQ  •  IF=  a  constant,     q.  e.  d. 


EXERCISES   V.  285 

137.  What  exception  does  136  suffer?  What  are  the 
relations  of  the  three  possible  cases  ? 

138.  Two  polar  conjugate  points  (or  rays)  and  the  pole 
(or  polar)  of  their  join  determine  a  polar  reciprocal  A. 

139.  Conjugate  points  on  a  ray  (or  rays  through  a  point) 
form  an  Involution.     When  positive?    When  negative ? 

140.  Two  conjugate  points  in  a  secant  divide  the  chord, 
and  two  conjugate  rays  through  a  point  divide  the  angle  of 
tangents  from  the  point,  harmonically. 

141.  The  outer  vertices  and  the  intersection  of  inner 
diagonals  of  an  encyclic  4-side  form  a  polar  A. 

142.  The  diagonals  of  a  pericyclic  4-side  form  a  polar  A. 

143.  Employ  141  and  142  to  find  the  polar  of  a  given 
pole  and  the  pole  of  a  given  polar  by  use  of  ruler  alone. 

144.  Given  a  centre  of  similitude  of  two  circles,  find  its 
polars  as  to  the  circles  and  the  power-axis  by  use  of  ruler 
alone. 

145.  Given  the  power-axis  of  two  circles,  find  its  poles 
as  to  the  circles  and  the  centres  of  similitude  by  use  of  ruler 
alone. 

Defs.  Two  figures  are  said  to  be  in  perspective  where 
the  joins  of  corresponding  points  all  go  through  a  point  called 
the  centre  of  projection.  — The  rays  are  called  rays  of  pro- 
jection. Parallel  rays  are  thought  concurrent  at  00.  — Two 
pencils  are  said  to  be  in  perspective  when  the  joins  of  cor- 
responding rays  all  lie  on  a  ray,  called  the  axis  of  projection. 

—  The  centre  of  projection  and  the  axis  of  projection  are 
plainly  self-correspondent.  —  Each  system  of  points  is  also 
said  to  be  in  perspective  with  the  pencil  of  rays  through  them. 

—  In  elementary  work  we  impose  the  condition  that  col- 
linear  points  in  the  one  figure  shall  correspond  to  collinear 
points  in  the  other. 


286  GEOMETRY. 

146.  Two  tracts  are  always  in  perspective  as  to  two  cen- 
tres. 

147.  Express  and  prove  the  reciprocal  theorem  as  to 
angles  (two-rays). 

148.  Three  collinear  points  and  three  concurrent  rays  are 
always  projective,  i.e.,  may  always  be  brought  mio  perspective. 

149.  Two  triplets  of  colHnear  points  are  always  in  pro- 
jection. (For  it  is  enough  to  slip  the  triplets  each  along  its 
ray  till  a  pair  of  correspondents  coincide.) 

150.  State  and  prove  the  reciprocal  theorem  for  pencils 
of  three. 

Def.  The  ratio  of  the  distances  of  any  third  ray  iV  of  a 
pencil  from  two  base-rays  L  and  M  of  the  pencil  is  called 
the  distance-ratio  of  the  third  ray  as  to  the  other  two ;  it 

may  be  written  ( )  and  is  reckoned  +  or  —  according 

\NM) 

as  the  angles  LN  and  NM  are  reckoned  in  the  same  or  in 
opposite  wise. 

151.  Trace  the  course  of  the  distance-ratio  as  the  third 
ray  completes  a  rotation  about  the  centre  of  the  pencil. 

152.  The  distance-ratio  equals  the  sine-ratio  of  the  angles 
formed  by  the  third  ray  with  the  base-rays. 

153.  When  two  distance-  or  sine-ratios  are  counter,  the 
four  rays  are  harmonic,  and  the  ratio  of  their  ratios  is  —  i. 

154.  Compare  distance-ratios  of  corresponding  angles 
and  tracts  in  a  pencil. 

Def.  The  ratio  of  two  distance-ratios  in  the  same  pencil, 
whether  of  tracts  or  of  angles,  is  called  the  cross-ratio,  or 
ratio  of  double  section  (^Doppelschnittsverhaeltniss),  or  an- 
harmonic  ratio,  of  the  bounding  points  or  rays,  and  is  written 
{A BCD)  or  {LMNF).     Its  value  is 

ABj_CD        sin  LM-  sin  NP 
BC'DA        sinMN'Sm^ 


EXERCISES    V.  287 

155.  The  joins  of  the  vertices  of  a  A  with  three  colHnear 
points  on  the  opposite  sides  divide  the  angles  so  that  the 
triple  sine-ratio  =  —  i . 

156.  Three  concurrent  rays  through  the  vertices  of  a  A 
divide  the  opposite  sides  so  that  the  triple  distance-ratio 
=  4-  I  {^Ceva,  1678). 

157.  Convert  these   two    theorems    and   those   of  Arts. 

158.  Apply  these  converses  in  establishing  the  concur- 
rences of  altitudes,  mid-normals,  mid-rays,  medials,  etc. 

159.  The  sides  of  a  A  cut  by  a  circle  in  six  points  are 
divided  so  that  the  continued  product  of  the  distance-ratios 
is  -f  I  {Carnot,  1 753-1823). 

160.  Express  and  prove  the  corresponding  proposition 
concerning  six  tangents,  drawn  from  three  points,  to  a  circle 
{ChasleSf  1850). 

Nint.  r  the  radius;  A,  B,  C  the  points;  AT^,  AT^  two 
tangents-lengths  =  /j,  4 ;  ^u  ^2  the  distances  of  T^,  T^  from 
BC ;  /i,  I2  the  intersections  of  two  tangents,  parallel  to  BC, 
with  the  rays  AT^,  AT2',  d,  e,  f  the  distances  of  BC,  CA, 
AB  from  the  centre  O ;  jPthe  projection  of  o  on  BC.     Then 

a^:  ti=^  d  —  r\  AIi,  a^,'.  ti=-  d  -\-  r  \  AI^, 

:,  aya.2 :  A4  =  d^  —  r^ :  AI^  •  AI^. 


But  from  similar  A  OAIi,  OAL  we  have  AI-  AI^  =  ACf ; 
:.  a^a^  \  tit^—d'—r^  \  A  O^.  Similarly,  byd2 :  tit2=^—r- :  AO'. 
.'.  a^ai :  ^A  —  d'^  —  r'^ :  e^  —  r^.  Find  two  analogous  equa- 
tions, combine  the  three  by  multiplication,  and  the  propo- 
sition in  question  results. 

161.  The  three  joins  of  the  opposite  sides  of  an  encyclic 
6-side  are  collinear  {Pascal,  1640). 

Hint.   Let.  i  2  3  4  5  6  be  the  6-side ;  I,  J,  K  the  joins 


288  GEOMETRY. 

of  the  opposite  pairs,  12  and  45,  23  and  56,  34  and  61. 
The  alternate  sides  61,  23,  45  form  a  A  ABC,  and  are  cut 
collinearly  by  the  other  alternate  sides  12,  34,  56;  apply 
thrice  the  theorem  of  Menelaos;  multiply,  cancel,  and  apply 
the  converse  of  the  theorem  of  Menelaos. 

162.  Express  and  prove  the  corresponding  theorem  of 
Brianchon  (1806),  using  Ceva's  theorem  and  converse. 

163.  Every  different  order  of  sides  respecting  vertices 
gives  a  different  hexagram  of  Pascal  respectively  hexagon  of 
Brianchon ;  how  many  of  each  are  possible  ? 

164.  These  so-called  Pascal  Y2,y^  are  concurrent,  and  the 
Brianchon  points  are  colhnear,  in  sets  of  three  {Steiner,  1832). 

165.  Apply  the  theorems  of  Pascal  and  Brianchon  to  find 
with  ruler  alone  a  tangent  to  a  given  circle  at  a  given  point 
and  the  point  of  touch  of  a  given  tangent  {Steiner,  1833). 

Def.  In  projecting  one  ray  L  on  another  Z',  there  will  be 
one  ray  of  projection/^r«//(?/to  L'  and  meeting  L  at  V.  To 
this  point  V,  and  to  it  only,  there  corresponds  no  finite  point 
of  Z' ;  to  points  close  at  will  to  V  there  correspond  points 
far  at  will  on  Z'.  Hence  V  is  called  the  vanishing  point  of 
Z  with  respect  to  Z'.  Similarly,  £/'  is  the  vanishing  point 
of  Z'  as  to  Z. 

166.  /*  and  Z^  correspond  on  Z  andZ';  show  and  state 
in  words  that  FV -  F'U'=  OF-  OU'. 

Def.  This  constant  rectangle  (product)  is  called  the 
constant  of  projection. 

167.  Two  A  are  always  projective.  (For  we  can  always 
place  a  pair  of  vertices  on  a  point,  or  a  pair  of  sides  on  a 
ray,  and  —  what  then  ?) 

168.  Three  pairs  of  points  taken  at  random  on  three  con- 
current rays,  each  pair  on  a  ray,  determine  two  perspective 
A  whose  corresponding  sides  meet  collinearly.  (Use  the 
propositions  of  Menelaos  and  Ceva.) 


EXERCISES    V.  289 

169.  Express  and  prove  the  reciprocal  of  168. 

1 70.  The  Locus  of  the  vanishing  points  of  all  rays  of  one 
of  two  (rectilinearly)  perspective  figures  is  a  ray  (called 
vanishing  ray)  parallel  to  the  Axis  of  projection. 

171.  The  distance  of  the  one  vanishing  ray  from  the  Axis 
equals  the  distance  of  the  other  from  the  Centre. 

172.  Parallel  rays  of  one  of  two  perspective  figures  corre- 
spond to  rays  concurrent  in  a  vanishing  point  of  the  other. 

173.  A  circle  is  in  perspective  with  itself,  any  pole  and 
corresponding  polar  being  Centre  and  Axis. 

1 74.  The  vanishing  ray  halves  the  distance  between  the 
Centre  and  the  Axis  and  is  the  Power-axis  of  the  circle  and 
the  Centre  of  projection  (regarded  as  a  point-circle). 

175.  Two  circles  are  in  perspective  as  to  a  centre  of 
similitude,  and  the  mid-parallel  of  the  polars  of  this  centre 
is  the  Axis. 


1 76.  A  figure  F  is  pushed  and  turned  about  in  a  plane 
into  any  other  position  P  \  show  that  the  same  change  of 
position  may  be  effected  by  simply  turning  about  a  point  in 
the  plane  called  the  Centre  of  Rotation. 

Hint.  A,  B,  C  three  points  of  F,  and  A\  B\  C  their 
positions  in  F ;  draw  the  mid-normals  ofAA',  BB\  CO ; 
etc. 

177.  If  the  joins  of  the  corresponding  vertices  of  two  A 
be  concurrent,  the  joins  of  the  corresponding  sides  are 
collinear;  and  conversely  (Desargues). 

1 78.  If  a  quadrilateral  be  inscribed  in  a  circle  Q  while 
two  of  its  opposite  sides  touch  a  circle  C^  and  the  other  two 
touch  a  circle  C3,  then  the  three  circles  Cj,  C^,  C^  are  co- 
axal, —  a  theorem  very  important  in  the  theory  of  Elliptic 
Functions. 


290  GEOMETRY. 

179.  The  area  of  a  pericyclic  polygon  equals  half  the 
rectangle  of  the  in-radius  and  the  perimeter. 

180.  If  i-  be  the  half-sum  of  the  sides  a,  b,  c  oi  2i  A,  and 
^1  ^15  ^2>  ^3  the  in-  and  ex-radii,  then  A  =  rs  =  ri{^s  —  a)  = 
r,{s-b)=r^{s-c). 

181.  Hence,  show  that  -  =  -  H 1 —  and  A^  =  rr^r^r^,. 

r      ri      r^      r^ 

182.  \i  a,b,c\yt  the  sides  of  a  A,  and  h  the  altitude  CO, 
show  from  a-  —  F^  -\-  r  —  2  c  -  AC  that 

/^2  =  54^V  -  (^2-f  c'~a^y\l4,c'  =  ^^s(s  -  a)  {s-  b)  {s-c), 

whence  A^  z=s{s  —  a)  {s  —  b)  {s  —  c)    {Hero,  250  B.C.) . 

1 83.  If  a,  b,  c  be  the  sides  of  a  A,  and  m  the  tract  from 
C  to  ^  halving  ^  C  and  cutting  c  into  parts  u  and  v,  show 

that  ab  —  Mv=  — ^ „  -  sis  —  c^. 

1 84.  If  two  such  tracts  in  a  A  be  equal,  the  A  is  sym- 
metric. 

1 8s.    Show  that  the  circum-radius  of  a  A  = . 

4A 

186.  Express  through  r  the  radius  of  a  circle,  the  sides 
and  areas  of  the  regular  inscribed  and  circumscribed  6-sides, 
4-sides,  3-sides,  lo-sides,  5 -sides;  also  the  apothegms  of  the 
in-polygons. 

187.  Given  the  centre  of  similitude  and  two  correspond- 
ing rays  of  two  similar  figures  in  perspective,  find  P^  corre- 
sponding to  a  given  P. 

188.  Corresponding  angles  of  similar  figures  in  perspec- 
tive have  always  the  same  sense. 

189.  If  the  A  ABC,  ABD,  etc.,  of  F  are  similar  to 
AB'C,  A'B'D\  etc.,  of  F\  then  7^  and  F'  are  similar. 

190.  Two  similar  figures  are  in  perspective  when  two 
corresponding  rays  are  parallel. 


EXERCISES    V.  291 

191.  Through  any  point  /*  (or -parallel  to  any  ray  iV), 
draw  a  ray  towards  the  inaccessible  intersection  of  the  rays 
L  and  M. 

192.  The  sides  of  a  quadrangle  are  given  in  position; 
draw  the  diagonals  when  two  opposite  vertices  are  inacces- 
sible, and  when  all  the  vertices  are  inaccessible. 

193.  Draw  a  circle  S^  in  perspective  with  S  as  to  the 
centre  O,  so  that  two  given  points /'and  /"  shall  correspond  ; 
so  that  two  given  parallels  L  and  V  shall  correspond ;  so 
that  S  shall  have  a  given  radius  r' ;  or,  so  that  the  centre  of 
S  shall  lie  on  a  given  ray. 

194.  Draw  S  tangent  to  .S  so  that  two  given  points  P 
and  P,  or  two  given  rays  L  and  L\  may  correspond. 

195.  Draw  a  circle  to  touch  a  given  circle  and  also  touch 
a  given  ray  at  a  given  point. 

196.  Draw  a  circle  tangent  to  two  given  rays  and  a  given 
circle. 

197.  In  any  pencil  of  four  rays,  as  OA,  OB,  OC,  OP 
—  written  0{ABCP)—AOB\  -  COP\^- BOC]- AOP\  + 
COA\'BOP\  =  o. 

Bint.  Note  that  AB 'p=  OA- OB •  AOB\,  etc.,  and 
use  66. 

198.  The  concurrent  rays  L,  M,  N  are  distant  /,  m,  n 
from  P;  prove  /•  MN\  -\- 7n  - NL\ -\- n - LM\  =  o. 


199.  If  2s  =  a-\-b-\-c-\-d=  perimeter  of  an  encyclic 
quadrangle,  show  that  A^  =  (^  —  a)  {s  —  b){s  —  c)  {s  —  d) 
and  express  this  result  symmetrically  through  a,  b,  c,  d. 

200.  If  r  and  r'  be  the  circum-radius  and  in-radius  and 
2s  =  a  -{-b  -\-  c  the  sum  of  the  sides  of  a  A,  prove  that 
2  rr^s  =  abc. 

201.  Draw  a  fourth  harmonic  to  three  rays  of  a  pencil. 


292  GEOMETRY, 

202.  The  sum  of  the  distances  of  the  sides  of  a  A  from  a 
point,  each  side  multiplied  by  the  sine  of  its  opposite  angle, 
is  constant  for  that  A. 

203.  A  ray  cuts  the  sides  a,  b,  c  oi  2,  /\  under  angles  a , 
P',  y' ;  show  that  a\' a'\ -{-^l- 13'\-\- y\'y'\  =  o. 

204.  Concurrent  rays  through  the  vertices  of  a  A  divide 
the  sides  so  that  the  continued  product  of  the  ratios  of 
division  is  —  i. 

205.  Concurrent  rays  through  the  vertices  A,  B,  C  of  a, 
A  cut  the  sides  at  P,  Q,  R ;  show  that  the  intersections 
/,  /,  K  of  AB  and  FQ,  BC  and  QR,  CA  and  RP  are 
collinear;  also  that'  if  BJ  and  CK,  CK  and  AI,  A I  and 
^/meet  in  S,  T,  U,  then  AS,  BT,  C^  concur. 


INDEX. 


(The  numerals  refer  to  pages.     Only  the  more  important  references  are  given.) 


Abstraction,  7. 

Addends,  17. 

Addition,  17. 

theorem,  248. 

Adjacent,  30. 

Alternation,  172. 

Alticentre,  68. 

Altitude,  67,  148. 

Ambiguous  case,  46. 

Angle,  19,  72, 

adjacent,  30. 

alternate,  54, 

complemental,  32. 

corresponding,  53. 

explemental,  31. 

interadjacent,  54. 

right,  31. 

round,  26. 

straight  or  flat,  26. 

supplemental,  31. 

vertical,  37. 

Anomaly,  191. 

Antecedents,  172. 

Anti-homologous,  217. 

Anti-parallelogram,  62,  65. 

Apollonius,  225. 

Apothem,  249. 

Arc,  13,  90. 

Area,  144. 

Areal  unit,  231. 

Arms,  19. 

Axal  symmetry,  79. 


Axally  symmetric,  77. 
Axis,  power  or  radical,  213. 

of  projection,  285. 

of  similitude,  216. 

of  symmetry,  jj. 

Babylonian,  70,  133. 

Band,  86. 

Base,  37,  148. 

Bases,  major  and  minor,  65. 

Beltrmni,  2j2. 

Bi-dimensional,  5. 

Bisect,  125. 

Bisector,  33. 

Bolyai,  269,  270. 

Border,  5. 

Boundary,  252. 

Boundless,  2. 

Brianchon,  288. 

Carnot,  287. 
Cell,  Peaucellier^s,  278. 
Central  symmetry,  yj. 
Centre,  18. 

■ of  circle,  11,  93. 

of  inversion,  168. 

of  involution,  284. 

of  pencil,  80. 

power-,  168,  213. 

of  projection,  285. 

of  symmetry,  77. 


293 


294 


GEOMETRY. 


Centric  figure,  215. 
Centroid,  66. 
Ceva,  287,  etc. 
Chasles,  287. 
Chord,  90,  283. 

of  contact,  108. 

Circle,  13,  90. 

of  inversion,  168. 

of  similitude,  282. 

Circum-circle  and  centre,  67,  96. 

Clifford,  274. 

Clockwise,    counter-clockwise,    25, 

71,  72. 
Closed,  72. 
Co-axal,  281. 
Collinear,  82. 
Common  point,  281. 
Commutative,  17. 
Compasses,  96,  192. 
Compendent,  145. 
Complanar,  25. 
Complement,  -al,  32,  96,  155. 
Compounded,  172. 
Conclusion,  25. 
Concur,  concurrent,  65. 
Congruent,  16,  passim. 
Conjugate,  93,  94,  222,  283. 
Consequence,  172. 
Continuous,  3. 
Contra-perspective,  188. 
Contra  positive,  40. 
Convex,  72. 
Corollary,  22. 

Correspond,  -ent,  16,  76,  145,  173. 
Cosine,  241. 
Cosines,  law  of,  245. 
Couplet,  172. 
Criteria,  147. 
Critical,  107. 
Crossed,  63. 
Cross-wise,  216. 
Cross-ratio,  286. 
Cuboid,  239. 


Dase,  253. 
Definites,  234. 
Definition,  59. 
Degrees,  70. 
Denominator,  228. 
Diagonal,  57,  63. 
Diameter,  93,  215,  283. 
Difference,  17,  172. 
Dimensions,  3,  149. 
Direct  perspective,  188. 
Dissimilarly,  219. 
Distance,  15,  20. 

ratio,  286. 

Divided,  172. 

,  similarly,  179. 

Division,  harmonic,  184. 

,  inner  and  outer,  182. 

Double,  78. 

Eidograph,  192. 
Ellipse,  94. 
Elliptic,  270,  271. 
Encyclic,  64,  passim. 
Enthymeme,  30. 
Envelope,  119. 
Equal,  -ity,  19,  147. 
Equator,  6. 
Equiangular,  70. 

,  mutually,  173. 

Equiareal,  267. 

Equidistant,  11. 

Equilateral,  70. 

Equivalent,  49. 

Erotetic,  27. 

Euclid,  -ian,  55,  159,  255,  271. 

Euler,  141. 

Even,  242. 

Ex-centre  and  circle,  69,  173. 

Explemental,  30,  95. 

Extreme  and  mean  ratio,  202. 

Extremes,  171. 

Family,  132. 


INDEX. 


295 


Fermat,  225. 
Feuerbach,  112. 
Figure,  16. 
Flat  angle,  26. 
Foci,  284. 
Four-side,  63. 
Fraction,  228, 
Frischauf,  270,  274. 
Function,  240,  242. 

Gaultier,  168. 
Gauss,  204,  271. 
Generated,  210. 
Geodetic,  269. 
Geometric  mean,  172. 
Gergonne,  225. 
Golden  section,  202. 

Half-strip,  87. 

Halsted,  269. 

Hankel,  234. 

Harmonic,  183,  2Z2,  passim. 

Helmholtz,  274. 

Henrici,  260. 

Heptagon,  116. 

Hero,  290. 

Hexagon,  hexagram,  288. 

Hippocrates,  276. 

Homoeoidal,  -ity,  2,  passim. 

nomothetic,  188. 

Hyperbola,  94. 

Hyperbolic,  270,  271. 

Hyper-euclidean,  55. 

Hyper-spaces,  274. 

Hypotenuse,  68. 

In-centreand  circle,  68. 
Incommensurable,  231. 
Infinite,  2,  252. 
Infinitesimal,  250. 
Inner,  innerly,  33,  63. 
Inscribed,  100. 
Instruments,  192. 


Inverse,  63,  216. 
Inverse  points,  168. 
Inversion,  216. 
Inverted,  172. 
Invertor,  278. 
Involution,  284. 
Irrational,  234. 
Isoclinal,  79. 
Isoperimetric,  264,  267. 
Isosceles,  37. 

Joins,  76. 

Kaleidoscope,  137. 
Killing-,  274. 
Kite,  83. 
Klein,  271. 
Kriimmungsmaass,  271. 

Lemma,  92. 

Length,  tangent-,  108,  213. 
Limit,  98,  252,  267. 
Limiting  points,  aSi. 
Line,  6. 
Linkage,  265. 
Lobaischevsky,  -an,  271. 
Locus,  12. 

Magnitudinal  unit,  228. 

Maximum  and  minimum,  165,  261. 

Means,  171. 

Measure  of  curvature,  271. 

Medial,  38. 

Median  section,  202. 

Menelaos,  239,  288. 

Metric  number,  228. 

M-fold,  226. 

Mid-normal,  37. 

Mid-parallel,  65. 

Mid-ray,  33. 

Minutes,  70. 

Montyon,  121. 

Multiple,  226. 


296 


GEOMETRY. 


N-angle,  72. 
Newcomb,  274. 
Nine-point  circle,  112. 
Non-intersectors,  54. 
5lormal,  31,  63,  102. 
Not-self,  273. 
N-side,  72. 
Numerator,  228. 
Numerics,  234. 

Odd,  245. 
Open,  72. 

Operation,  laws  of,  234. 
Origin,  71. 
Orthocentre,  68. 
Orthogonal,  108. 
Outer,  outerly,  33,  63. 

Pantagraph,  192. 
Pappus,  209. 
Parabolic,  270,  271. 
Parallel,  55. 
Parallelogram,  57. 
Pascal,  287. 
Peaucellier,  121,  278. 
Pencil,  80. 
Pergae,  225. 
Peri  cyclic,  117. 
Perimeter,  74,  115. 
Perimetric  ratio,  253. 
Period,  -ic,  -icity,  242. 
Peripheral,  periphery,  99. 
Permanence,  234. 
Perspective,  188,  285. 
Plane,  11. 

Polar,  pole,  108,  219,  etc. 
Polar  centre  and  circle,  281. 
Polygon,  71. 
Point,  6. 
Ponce  let,  281. 
Porism,  22. 
Postulate,  23,  122. 
Power,  166,  213. 


Power-axis  and  centre,  168,  213. 

Premisses,  29. 

Principle,  234. 

Problem,  120. 

Product,  235. 

Projection,  160,  286. 

,  constant  of,  288. 

Proportion,  etc.,  16^,  passim. 
Ptolemy,  180. 
Pythagoras,  159. 

Quadrilateral,  63. 

Radian,  255, 

Radical  axis  and  centre,  168. 

Radius,  95. 

vector,  2IO. 

Ratio,  181. 

of  double  section,  286. 

cross,  distance,  sine,  286. 

of  similitude,  213. 

Ray,  14. 

of  projection,  285. 

Reciprocity,  reciprocal,  80,  281. 

Rectangle,  52,  149. 

Reentrant,  72. 

Referee,  220. 

Regular,  70. 

Reversible,  reversibility,  ii,  79. 

Rhombus,  60. 

Rotation,  centre  of,  289. 

Riemann,  -ian,  11,  145,  271. 

Row,  80. 

Salmon,  281. 
Secant,  90. 
Seconds,  70. 
Sect,  16. 

Section,  ratio,  286. 
Sector,  95,  192. 
Segment,  95. 
Self-conjugate,  281. 

correspondent,  285. 

-reciprocal,  281. 


INDEX. 


297 


Semi-circle,  95. 

Seven-side,  116. 

Sexagesimal,  134. 

Sextant,  137. 

Shape,  56. 

Similar,  -ity,  77,  175,  188,  213. 

Similitude,  axis  and  centre  of,  213, 

216. 
Simson's  line,  279. 
Sine,  240. 
Sines,  law  of,  245. 
Sine-ratio,  286. 
Size,  56. 

Small  at  will,  230. 
Solid,  8. 
Space,  I. 

Spaces,  four  forms  of,  270. 
Sphere,  11. 
Spherics,  261. 
Square,  61,  157. 
Squaring  circle,  253. 
Steiner,  288. 
Strip,  86. 
Subtend,  90. 
Subtraction,  17. 
Sum,  17,  146,  172. 
Summand,  17,  146. 
Supplemental,  31,  96. 
Surface,  5. 
Surveying,  246. 


Symmetric,  77,  82. 

Symmetry,  axal  and  central,  76,  77. 

,  axis  and  centre  of,  'jt. 

System,  132. 

Taction-problem,  212. 
Tangent,  102. 

length,  108,  213. 

Terms,  171. 
Theorem,  22. 
Three-side,  87. 
Time- axis,  262. 
Tract,  16. 
Trapezoid,  65. 
Triangle,  35. 
Triangles,  similar,  56. 
Triply  laid,  87. 
Two-ray,  283. 

Unit-magnitude,  228. 

Vanishing  point  and  ray,  288,  289. 
Vertices,  35,  72. 
Vieta,  225. 

Westings,  246. 

Year,  70. 

Zweistrahl,  283. 


INTRODUCTORY  MODERN  GEOMETRY 

OF  THE 

POINT,  RAY,  AND  CIRCLE, 

BY 

WILLIAM   B.   SMITH,   Ph.D., 

Professor  of  Mathematics  in  Missouri  State  University. 


Now   Ready. 
Complete  edition,  $i.io. 

Copies  of  this  complete  edition  will  he  exchanged  for  such  copies  of 
Part  I.  (75  cents)  as  are  returned  to  the  publishers  in  good  condition, 
on  payment  of  the  difference  in  price. 


The  work  follows  the  lines  struck  out  by  the  great 
geometers  of  the  last  half  century,  and  presents  the 
subject  in  the  light  of  their  researches.  The  text  proper 
conducts  the  student  through  the  taction  problem  of 
Apollonius,  while  the  exercises  direct  him  much  further 
in  the  doctrines  of  perspective  and  projection. 

This  book  is  written  primarily  for  students  preparing 
for  admission  to  the  freshman  class  of  the  Missouri 
State  University,  and  has  already  been  thoroughly  tested 
in  the  sub-freshman  department  of  that  institution.  It 
covers  both  in  amount  and  quality  the  geometrical  in- 
struction required  for  admission  to  any  of  the  higher 
universities. 

MACMILLAN   &  CO., 

112  FOURTH  AVENUE,  NEW  YORK. 
1 


THE    PRINCIPLES 

OF 

ELEMENTARY  ALGEBRA, 

BY 

NATHAN   F.   DUPUIS,  M.A.,  F.R.S.C, 

Professor  of  Pure  Mathematics  in  the  University  of  Queen's  College,  Kingston, 
Canada. 

i2mo.    $1.10. 


FROM  THE  AUTHOR'S  PREFACE. 

The  whole  covers  pretty  well  the  whole  range  of  elementary  algebraic 
subjects,  and  in  the  treatment  of  these  subjects  fundamental  principles 
and  clear  ideas  are  considered  as  of  more  importance  than  mere  mechan- 
ical processes.  The  treatment,  especially  in  the  higher  parts,  is  not 
exhaustive;  but  it  is  hoped  that  the  treatment  is  sufliciently  full  to 
enable  the  reader  who  has  mastered  the  work  as  here  presented,  to  take 
up  with  profit  special  treatises  upon  the  various  subjects. 

Much  prominence  is  given  to  the  formal  laws  of  Algebra  and  to  the 
subject  of  factoring,  and  the  theory  of  the  solution  of  the  quadratic  and 
other  equations  is  deduced  from  the  principles  of  factorization. 

OPINIONS  OF   TEACHERS. 

"  It  approaches  more  nearly  the  ideal  Algebra  than  any  other  text- 
book on  the  subject  I  am  acquainted  with.  It  is  up  to  the  time,  and 
lays  stress  on  those  points  that  are  especially  important."  —  Prof.  W. 
P.  DuRFEE,  Hobart  College,  N.Y. 

"It  is  certainly  well  and  clearly  written,  and  I  can  see  great  advan- 
tage from  the  early  use  of  the  Sigma  Notation,  Synthetic  Division,  the 
Graphical  Determinants,  and  other  features  of  the  work.  The  topics 
seem  to  me  set  in  the  proper  proportion,  and  the  examples  a  good  selec- 
tion."—  Prof.  E.  P.  Thompson,  Westminster  College,  Pa. 

"  I  regard  this  as  a  very  valuable  contribution  to  our  educational 
literature.  The  author  has  attempted  to  evolve,  logically,  and  in  all  its 
generality,  the  science  of  Algebra  from  a  few  elementary  principles 
(including  that  of  the  permanence  of  equivalent  forms) ;  and  in  this  I 
think  he  has  succeeded.  I  commend  the  work  to  all  teachers  of  Algebra 
as  a  science."  — Prof.  C.  H.  Judson,  Furman  University,  S.C. 


MACMILLAN   &  CO., 

112  FOURTH  AVENUE,  NEW  YORK. 

2 


MATHEMATICAL  WORKS 

BY 

CHARLES  SMITH,  M.A., 

Master  of  Sidney  Sussex  College,  Cambridge. 


A  TREATISE  ON  ALGEBRA. 

New  and  enlarged  edition  now  ready. 

ISmo.    $1.90. 

^*^  This  new  edition  has  been  greatly  improved  by  the  addi- 
tion of  a  chapter  on  Differential  Equations,  and  other  changes. 

No  better  testimony  to  the  value  of  Mr.  Smith's  work  can  be 
given  than  its  adoption  as  the  prescribed  text-book  in  the  follow- 
ing Schools  and  Colleges,  among  others  :  — 

University  of  Michigan,  Ann  Arbor,  Mich. 

University  of  Wisconsin,  Madison,  Wis. 

Cornell  University,  Ithaca,  N.Y. 
University  of  Pennsylvania,  Philadelphia,  Pa. 

University  of  Missouri,  Columbia,  Mo. 

Washington  University,  St.  Louis,  Mo. 
University  of  Indiana,  Bloomington,  Ind. 

Bryn  Mawr  College,  Bryn  Mawr,  Pa. 

Rose  Polytechnic  Institute,  Terre  Haute,  Ind. 
Chicago  Manual  Training  School,  Chicago,  111. 

Leland  Stanford  Jr.  University,  Palo  Alto,  Cal. 

Michigan  State  Normal  School,  Ypsilanti,  Mich. 
Etc.    Etc.    Etc. 
Key,  sold  only  on  the  written  order  of  a  teacher,  §2.60. 


MACMILLAN  &  CO., 

112  FOURTH  AVENUE,  NEW  YORK. 
3 


ELEMENTARY  ALGEBRA. 

By  Charles  Smith,  M.A.,  Master  of  Sidney  Sussex  College,  Cam- 
bridge. Second  edition,  revised  and  enlarged,  pp.  viii,  404.  16mo. 
$1.10. 


FROM  THE  AUTHOR'S  PREFACE. 

"  The  whole  book  has  been  thoroughly  revised,  and  the  early  chapters 
remodelled  and  simplified;  the  number  of  examples  has  been  very 
greatly  increased ;  and  chapters  on  Logarithms  and  Scales  of  Notation 
have  been  added.  It  is  hoped  that  the  changes  which  have  been  made 
will  increase  the  usefulness  of  the  work." 

From  Prof.  J.  P.  NAYLOR,  of  Indiana  University. 

"  I  consider  it,  without  exception,  the  best  Elementary  Algebra  that 
I  have  seen." 

PRESS    NOTICES. 

"The  examples  are  numerous,  well  selected,  and  carefully  arranged. 
The  volume  has  many  good  features  in  its  pages,  and  beginners  will 
find  the  subject  thoroughly  placed  before  them,  and  the  road  through 
the  science  made  easy  in  no  small  degree."  —  Schoolmaster. 

"There  is  a  logical  clearness  about  his  expositions  and  the  order  of 
his  chapters  for  which  schoolboys  and  schoolmasters  should  be,  and 
will  be,  very  grateful."  —  Educational  limes. 

"  It  is  scientific  in  exposition,  and  is  always  very  precise  and  sound. 
Great  pains  have  been  taken  with  every  detail  of  the  work  by  a  perfect 
master  of  the  subject."  —  School  Board  Chronicle. 

"This  Elementary  Algebra  treats  the  subject  up  to  the  binomial 
theorem  for  a  positive  integral  exponent,  and  so  far  as  it  goes  deserves 
the  highest  commendation."  —  Athenseum. 

"  One  could  hardly  desire  a  better  beginning  on  the  subject  which  it 
treats  than  Mr.  Charles  Smith's  '  Elementary  Algebra.'  .  . .  The  author 
certainly  has  acquired  —  unless  it  'growed'  —  the  knack  of  writing, 
text-books  which  are  not  only  easily  understood  by  the  junior  student, 
but  which  also  commend  themselves  to  the  admiration  of  more 
matured  ones."  —  Saturday  Review. 


MACMILLAN   &  CO., 

112  FOURTH  AVENUE,  NEW  YORK. 
4 


A  PROGRESSIVE  SERIES  ON  ALGEBRA 

BY 
MESSRS.  HALL  &  KNIGHT. 


Algebra  for  Beginners.     By  H.  S.  Hall,  M.A.,  and  S.  K. 
Knight,  B.A.     16mo.     Cloth.     60  cents. 

FROM  THE  AUTHOR'S  PREFACE. 

**  The  present  work  has  been  undertaken  in  order  to  supply  a  demand 
for  an  easy  introduction  to  the  'Elementary  Algebra  for  Schools/  and 
also  meet  the  wishes  of  those  who,  while  approving  of  the  order  and 
treatment  of  the  subject  there  laid  down,  have  felt  the  want  of  a  be- 
ginners' text-book  in  a  cheaper  form." 


Elementary  Algebra  for  Schools.  By  H.  S.  Hall,  M. A., 
and  S.  R.  Knight,  B.A.  16mo.  Cloth.  Without  an- 
swers, 90  cents.     With  answers,  $1.10. 

NOTICES  OF   THE   PRESS. 

"  This  is,  in  our  opinion,  the  best  Elementary  Algebra  for  school  use. 
It  is  the  combined  work  of  two  teachers  who  have  had  considerable  ex- 
perience of  actual  school  teaching,  .  .  .  and  so  successfully  grapples 
with  difficulties  which  our  present  text-books  in  use,  from  their  authors 
lacking  such  experience,  ignore  or  slightly  touch  upon.  .  .  .  We  con- 
fidently recommend  it  to  mathematical  teachers,who,  we  feel  sure,  will 
find  it  the  best  book  of  its  kind  for  teaching  purposes."  —  Nature. 

"  We  will  not  say  that  this  is  the  best  Elementary  Algebra  for  school 
use  that  we  have  come  across,  but  we  can  say  that  we  do  not  remember 
to  have  seen  a  better.  ...  It  is  the  outcome  of  a  long  experience  of 
school  teaching,  and  so  is  a  thoroughly  practical  book.  All  others  that 
we  have  in  our  eye  are  the  works  of  men  who  have  had  considerable 
experience  with  senior  and  junior  students  at  the  universities,  but  have 
had  little  if  any  acquaintance  with  the  poor  creatures  wlio  are  just 
stumbling  over  the  threshold  of  Algebra.  .  .  .  Buy  or  borrow  the  book 
for  yourselves  and  judge,  or  write  a  better.  ...  A  higher  text-book  is 
on  its  way.  This  occupies  sufficient  ground  for  the  generality  of  boys." 
—  Academy. 

MACMILLAN   &  CO., 

112  FOUETH  AVENUE,   NEW  YORK. 


HIGHER  ALGEBRA. 

A  Sequel  to  Elementary  Algebra  for  Schools.  By  H.  S. 
Hall,  M.A.,  and  S.  R.  Knight,  B.A.  Fourth  edition, 
containing  a  collection  of  three  hundred  Miscellaneous 
Examples,  which  will  be  found  useful  for  advanced 
students.     12mo.     $1.90. 


OPINIONS  OF   THE  PRESS. 

"  The  *  Elementary  Algebra  '  by  the  same  authors,  which  has  already 
reached  a  sixth  edition,  is  a  work  of  such  exceptional  merit  that  those 
acquainted  with  it  will  form  high  expectations  of  the  sequel  to  it  now 
issued.  Nor  will  they  be  disappointed.  Of  the  authors'  '  Higher  Alge- 
bra,' as  of  their  'Elementary  Algebra,'  we  unhesitatingly  assert  that 
it  is  by  far  the  best  work  of  its  kind  with  which  we  are  acquainted.  It 
supplies  a  want  much  felt  by  teachers."  —  The  Athenseum. 

"...  It  is  admirably  adapted  for  college  students,  as  its  predecessor 
was  for  schools.  It  is  a  well-arranged  and  well-reasoned-out  treatise, 
and  contains  much  that  we  have  not  met  with  before  in  similar  works. 
For  instance,  we  note  as  specially  good  the  articles  on  Convergency  and 
Divergency  of  Series,  on  the  treatment  of  Series  generally,  and  the 
treatment  of  Continued  Fractions.  .  .  .  The  book  is  almost  indispensa- 
ble, and  will  be  found  to  improve  upon  acquaintance."  —  The  Academy. 

"  We  have  no  hesitation  in  saying  that,  in  our  opinion,  it  is  one  of 
the  best  books  that  have  been  published  on  the  subject.  .  .  .  The  last 
chapter  supplies  a  most  excellent  introduction  to  the  Theory  of  Equa- 
tions. We  would  also  specially  mention  the  chapter  on  Determinants  and 
their  application,  forming  a  useful  preparation  for  the  reading  of  some 
separate  work  on  the  subject.  The  authors  have  certainly  added  to 
their  already  high  reputation  as  writers  of  mathematical  text-books  by 
the  work  now  under  notice,  which  is  remarkable  for  clearness,  accu- 
racy, and  thoroughness.  .  .  .  Although  we  have  referred  to  it  on  many 
points,  in  no  single  instance  have  we  found  it  wanting."  —  The  School 
Guardian. 


MACMILLAN  &  CO., 

112  FOURTH  AVENUE,  NEW  YORK. 


WORKS  BY  THE  REV.  J.  B.  LOCK, 

FELLOW  AND  BURSAR  OF  GONVILLE  AND   CAIUS  COLLEGE,  CAMBRIDGE. 
FORMERLY  MASTER  AT   ETON. 


Arithmetic  for  Schools.  3d  edition,  revised.  Adapted  to 
American  Schools  by  Prof.  Charlotte  A.  Scott,  Bryn 
Mawr  College,  Pa.     70  cents. 

"  Arithmetic  for  Schools,  by  the  Rev.  J.  B.  Lock,  is  one  of  those 
works  of  which  we  have  before  noticed  excellent  examples,  written  by 
men  who  have  acquired  their  power  of  presenting  mathematical  subjects 
in  a  clear  light  to  boys  by  actual  teaching  experience  in  schools.  Of  all 
the  works  which  our  author  has  now  written,  we  are  inclined  to  think 
this  the  best."  —  Academy. 

Trigonometry  for  Beginners,  as  far  as  the  Solution  of  Tri- 
angles.    3d  edition.     75  cents. 

"  It  is  exactly  the  book  to  place  in  the  hands  of  beginners."—  The 
Schoolmaster. 

Key  to  the  above,  supplied  on  a  teacher's  order  only,  $1.75. 
Elementary  Trigonometry,  with  chapters  on  Logarithms  and 

Notation.    0th  edition,  carefully  revised.    $1.10.    Key,  $1.75. 

"  The  work  is  carefully  and  intelligently  compiled."— 2%e  Athenssum. 

Trigonometry  of  One  Angle,  intended  for  those  students  who 

require  a  knowledge  of  the  properties  of  "sines  and  cosines  " 

for  use  in  the  study  of  elementary  mechanics.     65  cents. 

A  Treatise  on  Higher  Trigonometry.     3d  edition.    $1.00. 

"  Of  Mr.  Lock's  Higher  Trigonometry  we  can  speak  in  terms  of  un- 
qualified praise.  ...  In  conclusion,  we  congratulate  Mr.  Lock  upon  the 
completion  of  his  task,  which  enables  both  teachers  and  students  to 
keep  up  with  the  progress  made  of  late  years,  particularly  in  the  higher 
parts  of  this  branch  of  mathematics ;  and  we  regard  the  entire  series 
as  a  most  valuable  addition  to  the  text-books  on  the  subject." — Engi- 
neering. 

Elementary  and  Higher  Trigonometry,  the  Two  Parts  in  One 

Volume.     $1.90. 
Dynamics  for  Beginners.    $1.00. 

"  This  is  beyond  all  doubt  the  most  satisfactory  treatise  on  Elemen- 
tary Dynamics  that  has  yet  appeared."  — Engineering. 

Elementary  Statics.    $1.10. 

"This  volume  on  statics  .  .  .  is  admirable  for  its  careful  gradations 
and  sensible  arrangement  and  variety  of  problems  to  test  one's  knowl- 
edge of  that  subject."  —  The  Schoolmaster. 

Mechanics  for  Beginners.     Part  I.     90  cents. 

Euclid  for  Beginners.     Book  I.     60  cents. 

MACMILLAN   &   CO., 

112  FOURTH   AVENUE,   NEW   YORK. 

7 


WORKS   ON  TRIGONOMETRY 

PUBLISHED   BY 

MACMILLAN  &  CO. 


BOTTOMLEY.  —  Four  Figure  Mathematical  Tables.  Com- 
prising Logarithmic  and  Trigonometrical  Tables,  and  Tables 
of  Squares,  Square  Root,  and  Reciprocals.  By  J.  T.  Bot- 
TOMLEY,  M.A.,  F.R.G.S.,  F.C.S.     8vo.     70  cents. 

DYER  and  WHITCOMBE.  —The  Elements  of  Trigonometry. 
By  J.  M.  Dyer,  M.A.,  and  the  Rev.  R.  H.  Whitcombe, 
M.A.     11.25. 

HOBSON.  — A  Treatise  on  Plane  Trigonometry.  By  E.  W. 
HoBSON,  Sc.D.     8vo.     13.00. 

HOBSON  and  JESSOP.  —  An  Elementary  Treatise  on  Plane 
Trigonometry.  By  E.  W.  Hobson,  Sc.D.,  and  C.  M.  Jessop, 
M.A.     !^1.25. 

JOHNSON.  —  Treatise  on  Trigonometry.  By  W.  E.  Johnson, 
M.A. ,  formerly  Scholar  of  King's  College,  Cambridge.  12mo. 
12.25. 

LEVETT  and  DAVISON.  —  The  Elements  of  Trigonometry. 
By  Rawdox  I^evett  and  A.  F.  Davison,  Masters  at  ICing 
Edward's  School,  Birmingham.     Crown  8vo.     ^1.60. 

This  book  is  intended  to  be  a  very  easy  one  for  beginners,  all  diffi- 
culties connected  with  the  application  of  algebraic  signs  to  geometry, 
and  with  the  circular  measure  of  angles  being  exchided  from  Part  I. 
Part  II.  deals  with  the  real  algebraical  quantity,  and  gives  a  fairly  com- 
plete treatment  and  theory  of  the  circular  and  hyperbolic  functions 
considered  geometrically.  In  Part  III.  complex  numbers  are  dealt  with 
geometrically,  and  the  writers  have  tried  to  present  much  of  De  Mor- 
gan's teaching  in  as  simple  a  form  as  possible. 

WORKS   BY   THE   REV.   J.  B.   I.OCK. 

LOCK.  —  Trigonometry  for  Beginners.     As  far  as  the  Solution 

of  Triangles.     16mo.     75  cents.     Key,  $1.75. 

*'  A  very  concise  and  complete  little  treatise  on  this  somewhat  diffi- 
cult subject  for  boys ;  not  too  childishly  simple  in  its  explanations ;  an 
incentive  to  thinking,  not  a  substitute  for  it.  The  schoolboy  is  encour- 
aged, not  insulted.  The  illustrations  are  clear.  Abundant  examples 
are  given  at  every  stage,  with  answers  at  the  end  of  the  book,  the  gen- 
eral correctness  of  which  we  have  taken  pains  to  prove.  The  definitions 
are  good,  the  arrangement  of  the  work  clear  and  easy,  the  book  itself 
well  printed.  The  introduction  of  logarithmic  tables  from  one  hundred 
to  one  thousand,  with  explanations  and  illustrations  of  their  use,  espe- 
cially in  their  application  to  the  measurement  of  heights  and  dis- 
tances, is  a  very  great  advantage,  and  affords  opportunity  for  much 
useful  exercise."  — Journal  of  Education. 

Trigonometry  of  One  Angle.     Intended  for  those  students  who 
require  a  knowledge  of  the  properties  of  "  sines  and  cosines  " 
for  use  in  the  study  of  elementary  mechanics.     65  cents. 
8 


Elementary  Trigonometry.     6th  edition.     (In  this  edition  the 
chapter  on  Logarithms  has  been  carefully  revised.)     16mo. 
$1.10.     Key,  r2.25. 
"  The  work  contains  a  very  large  collection  of  good  (and  not  too  bard) 
examples.    Mr.  Lock  is  to  be  congratulated,  when  so  many  Trigonome- 
tries are  in  the  tield,  on  having  produced  so  good  a  book;  for  he  has 
not  merely  availed  himself  of  the  labors  of  his  predecessors,  but  by  the 
treatment  of  a  well-worn  subject  has  invested  the  study  of  it  with  in- 
terest." —  Nature. 

Eiigineerinr/ sa.ys :  "Mr.  Lock  has  contrived  to  invest  his  subject 
with  freshness.  His  treatment  of  circular  measure  is  very  clear,  and 
calculated  to  give  a  beginner  clear  ideas  respecting  it.  Throughout  the 
book  we  notice  neat  geometrical  proofs  of  the  various  theorems,  and  the 
ambiguous  case  is  made  very  clear  by  the  aid  of  both  geometry  and 
analysis.  The  examples  are  numerous  and  interesting,  and  the  methods 
used  in  working  out  those  which  are  given  as  illustrations  are  terse  and 
instructive." 

Higher  Trigonometry.     5th  edition.     16mo.    $1.00. 
Elementary  and  Higher  Trigonometry  in  one  vol.     <$1.90. 
McClelland   and   PRESTON.  —  a  Treatise  on  Spherical 
Trigonometry.     With  applications  to  Spherical  Geometry, 
and  numerous  examples.     By  William  J.  McClelland, 
M.A.,  and  Thomas  Preston,  B.A.    12mo.    Part  I.    $1.10. 
Part  II.     $1.25.     Two  Parts  in  one  volume,  $2.25. 
Ought  to  fill  an  important  gap  in  our  mathematical  libraries,  es- 
pecially as  there  are  many  sets  of  selected  examples,  with  hints  for 
solution.  —  Saturday  Eevieio. 
NIXON.  —  Elementary   Plane    Trigonometry.      By  R.  C.  J. 

Nixon,  M.A.     16mo.     $1.90. 
PALMER. —Practical  Logarithms  and  Trigonometry,  Text- 
Book  of.     By  J.  H.  Palmer.     16mo.     $1.10. 
TODHUNTER.  —  Trigonometry   for  Beginners.      By   Isaac 

ToDHUNTER,  F.R.S.     18mo.     60  cts.     Key,  $2.25. 
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Fellow  of  St.  John's  College,  Cambridge.     12mo.     $1.10. 
VYVYAN.  —  Introduction  to  Plane  Trigonometry.    By  the  Rev. 
T.  G.  Vyvyan,  M.A.     3d  ed.,  revised  and  corrected.     90  cts. 
WARD.  — Trigonometry  Examination  Papers.     60  cents. 
WOLSTENHOLME.  — Examples  for  Practice  in  the  Use  of 
Seven-Figure  Logarithms.      By  Joseph   Wolstenholme, 
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i,t%  Keys  are  sold  only  upon  a  teacher's  written  order. 


MACMILLAN  &  CO.,  112  Fourth  Avenue,  New  York. 


ELEMENTARY  SYNTHETIC  GEOMETRY 

OF   THE 

POINT,   LINE,  AND  CIRCLE  IN   THE  PLANE. 

By  Nathan  F.  Dupuis,  M.A.,  F.R.C.S.,  Professor  of  Mathematics  in 
Queen's  College,  Kingston,  Canada.    16mo.    |;1.10. 


FROM  THE  AUTHOR'S  PRKFACE. 

"  The  present  work  is  a  result  of  the  author's  experience  in  teaching 
geometry  to  junior  classes  in  the  University  for  a  series  of  years.  It 
is  not  an  edition  of  'Euclid's  Elements,'  and  has  in  fact  little  relation 
to  that  celebrated  ancient  work  except  in  the  subject-matter. 

"An  endeavor  is  made  to  connect  geometry  with  algebraic  forms 
and  symbols  :  (1)  by  an  elementary  study  of  the  modes  of  representative 
geometric  ideas  in  the  symbols  of  algebra ;  and  (2)  by  determining  the 
consequent  geometric  interpretation  which  is  to  be  given  to  each  inter- 
pretable  algebraic  form.  ...  In  the  earlier  parts  of  the  work  Con- 
structive Geometry  is  separated  from  Descriptive  Geometry,  and  short 
descriptions  are  given  of  the  more  important  geometric  drawing  instru- 
ments, having  special  reference  to  the  geometric  principle  of  their 
actions. . . .  Throughout  the  whole  work  modern  terminology  and 
modern  processes  have  been  used  with  the  greatest  freedom,  regard 
being  had  in  all  cases  to  perspicuity. .  . . 

"  The  whole  intention  in  preparing  the  work  has  been  to  furnish  the 
student  with  the  kind  of  geometric  knowledge  which  may  enable  him 
to  take  up  most  successfully  the  modern  works  on  analytical  geom- 
etry." 

"  To  this  valuable  work  we  previously  directed  special  attention.  The 
whole  intention  of  the  work  is  to  prepare  the  student  to  take  up  suc- 
cessfully the  modern  works  on  analytical  geometry.  It  is  safe  to  say 
that  a  student  will  learn  more  of  the  science  from  this  book  in  one 
year  than  he  can  learn  from  the  old-fashioned  translations  of  a  certain 
ancient  Greek  treatise  in  two  years.  Every  mathematical  master 
should  study  this  book  in  order  to  learn  the  logical  method  of  present- 
ing the  subject  to  beginners."  —  Canada  Educational  Journal. 


MACMILLAN   &   CO., 

112   FOUETH  AVENUE,    NEW   YORK. 
10 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

This  book  is  DUE  on  the  last  date  stamped  below. 

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